Revision as of 13:03, 24 January 2018

Abstract

Este artigo apresenta um novo esquema explícito para a solução das equações de águas rasas em uma e duas dimensões, desenvolvido a partir do método dos elementos de conservação e elementos de solução espaço-tempo, aqui abreviado por método CE/SE. As funções de base utilizadas são expansões de Taylor de segunda ordem no tempo e no espaço. Esse aumento na ordem das funções de aproximação produz o aumento no número de variáveis de marcha no tempo, por isso, além das variáveis fluxo e suas inclinações, também são incógnitas no presente esquema suas derivadas espaciais de segunda ordem. Um processo iterativo para o cálculo das derivadas de primeira e segunda ordem é formulado para problemas com choques e descontinuidades. Experimentos computacionais demonstram acurácia de terceira ordem. Os problemas de ruptura de barragem unidimensional e bidimensional considerados validam a acurácia e robustez do esquema.

keywords

Leis de conservação, Equações de águas rasas, Volume de controle espaço-tempo, Alta ordem de acurácia.

High-Order CE/SE Scheme for Dam-Break Flow Simulation

Abstract

This paper presents a new explicit scheme for the solution of shallow water equations in one and two space dimensions, developed from the space-time conservation element and solution element (CE/SE) method. The basis functions used are second-order Taylor expansions in time and space. This increase in the order of the approximation functions produces an increase in the number of unknowns in the scheme, therefore, besides the flow variables and their slopes, their second-order partial derivatives are also unknown in the present scheme. An iterative process for the calculation of the first and second order derivatives is formulated for problems with shocks and discontinuities. Computational experiments demonstrate third-order accuracy. The one-dimensional and two-dimensional dam-break problems presented validate the accuracy and robustness of this scheme.

Keywords: Conservation laws, Shallow water equations, Space-time control volume, High-order accuracy.

3.1 Introduco

As equações de águas rasas constituem um dos modelos mais comumente usados na análise de fluxo de água em rios ou áreas costais [1], sobretudo no que tange à simulações de fluxo de ruptura de barragem [2,3,4,5], parte essencial do projeto e avaliação da segurança de barragens, controle de inundação de rios, mitigação de desastres de bacias hidrográficas, etc [6]. Além da importância prática em hidráulica e engenharia costal, fornecem um ótimo modelo matemático para equações diferenciais hiperbólicas não lineares que podem ter soluções como ondas de choque [7] e, por isso, também são frequentemente utilizadas como problema de referência para novos esquemas de previsão numérica [8,9].

Extensivos estudos numéricos foram feitos no sentido de analisar os fenômenos governados por essas equações e diversos métodos foram desenvolvidos, como de volumes finitos [10,11,12], elementos finitos [13,14,15,16], diferenças finitas [17,18] e outros [19,20].

Recentemente, um novo método baseado em volumes de controle, denominado método dos elementos de conservação e elementos de solução espaço-tempo, do inglês, space-time conservation element and solution element method (CE/SE), foi proposto por Chang e To (1991) [21] e aplicações do mesmo em problemas de ruptura de barragem logo foram feitas [22,23,24]. Este método foi desenvolvido com o objetivo de obter soluções numéricas de leis de conservação e possui características importantes [25,26,27,28]: sua construção combina informações de ambas as formas diferencial e integral das leis de conservação; o conjunto de variáveis de marcha é formado pelas variáveis fluxo e suas derivadas espaciais; não são utilizadas técnicas de interpolação ou extrapolação nos valores da malha; não são utilizadas técnicas baseadas nas características, o que torna o método de fácil extensão à várias dimensões, bem como aplicável à equações não hiperbólicas, como as equações de Navier-Stokes, por exemplo; possui uma simples notação estêncil, o que facilita a programação. É caracterizado, também, por sua versatilidade, tendo em vista que pode ser adaptado à malhas não uniformes e não estruturadas [29,30], bem como pode ser estendido à altas ordens [31,32], razão pela qual possui aplicações nas mais diversas áreas [33,34,35,36].

O objetivo deste trabalho é apresentar a construção de um novo esquema CE/SE explícito de alta ordem para as equações de águas rasas e verificar sua eficiência em problemas de ruptura de barragem. Para isso, este artigo está organizado da seguinte maneira: na seção 3.2 é apresentado o modelo matemático a ser estudado (subseção 3.2.1), bem como é desenvolvido o esquema numérico discreto (subseção 3.2.2); na seção 3.3, experimentos numéricos unidimensionais e bidimensionais são considerados com a finalidade de validar o método; as conclusões são tecidas junto à seção 3.4 e, ao final, na seção 3.5, dispõe-se a lista de referências.

3.2 Equações Governantes e Método CE/SE

3.2.1 Equações de Águas Rasas

As equações de águas rasas, também conhecidas como equações de Saint Venant, possuem a seguinte forma vetorial conservativa

em que

sendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h}

a profundidade do fluxo (m); Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v}
são as velocidades médias  do escoamento (m/s) nas direções Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y}

, respectivamente; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g}

é a aceleração da gravidade (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 9.812}

m/sFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^2} ); Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_{0{x}} = \partial Z_b/\partial x}

 e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_{0{y}} = \partial Z_b/ \partial y}
são as inclinações do fundo do  canal cuja topografia é dada por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z_b}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_{f{x}}}

 e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_{f{y}}}
são os termos de resistência ao escoamento,  nas direções Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y}

, respectivamente, definidos por

em que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

(s/mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^{1/3}}

) é o coeficiente de fricção de Manning.

3.2.2 Esquema CE/SE

Considere, por simplicidade, a Eq. (3.1) como

então, pelo teorema da divergência no espaço Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}^3} , tem-se que a Eq. (3.5) representa a forma diferencial da lei integral de conservação

em que HFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle _m = (f_m, g_m,q_m)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S(V)}
representa o contorno de uma região  espaço-tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V \subset \mathbb{R}^3}

.

No método CE/SE, um elemento de conservação (CE) é uma região espaço-tempo em que a conservação do fluxo, Eq. (3.6), é forçada, enquanto que um elemento de solução (SE) é uma região espaço-tempo, normalmente distinta, em que as variáveis fluxo são supostamente suaves e a Eq. (3.5) é válida. Para definir estes dois importante objetos, considere primeiramente uma malha no plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y} , conforme Fig. ma. Existem dois grupos de pontos, marcados por círculos e cruzes, que representam nós da malha em dois níveis de tempo diferentes. A cada ponto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (i,j,k)}

da malha associa-se um CE e um SE. O CE é definido como o quadrilátero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle EFGHE'F'G'H'}
e o SE é a união do quadrilátero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P''Q''R''S''P'Q'R'S'}
e o polígono Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle EFGH}
(veja  Fig. 1a).

e (b) as definições dos  elementos de solução SEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (i,j,k)
e elementos de conservação CEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (i,j,k)
no ponto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (i,j,k)
da malha  [2012_Zhang, 1999_Zhang].

Para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (x,y,t) \in \mbox{SE}(i,j,k)} , aproxima-se Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_m}

por um polinômio de Taylor de segunda ordem

Aproximações análogas são feitas sobre as funções Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_m}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g_m}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3} . O termo fonte, por outro lado, é aproximado por uma ordem a menos:

Dessa forma, a Eq. (3.5) pode ser aproximada no SEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (i,j,k)}

por div HFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle _m^* = S_m^*}

, em que HFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle _m^* = \left(f_m^*, g_m^*, q_m^*\right)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3} , ao mesmo tempo em que a Eq. (3.6) é aproximada no CEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (i,j,k)}

por

3.2.2.1 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (q_m)_{i,j}^k

Substituindo as funções Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_m^*, f_m^*, g_m^*}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_m^*}
em (3.9), após diversas simplificações,  obtém-se o esquema de avanço no tempo para a variável Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{i,j}^k}

em que

É importante notar que as Equações (3.10) e (3.11) dependem apenas das incógnitas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_m, (q_m)_x, (q_m)_y, (q_m)_{xx}, (q_m)_{yy}}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{xy}}
do tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_{k-1/2}}

, para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3}

 pois, pelas Equações (3.1)-(3.4), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_m}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g_m}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_m}
também dependem destas últimas.  Por  outro lado, é preciso conhecer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{xx}]_{i,j}^k}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{yy}]_{i,j}^k}
previamente à obtenção de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{i,j}^k}
no tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_k}

, conforme Eq. (3.10). A seção a seguir descreve a obtenção dessas derivadas espaciais duplas.

3.2.2.2 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{xx}]_{i,j}^k , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{xy}]_{i,j}^k e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{yy}]_{i,j}^k 3.2.2.2 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q m) {xx}] {i,j}^k , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q m) {xy}] {i,j}^k e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q m) {yy}] {i,j}^k

Para prosseguir com o cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{xx}]_{i,j}^k} , constrói-se primeiramente uma equação auxiliar a partir da Eq. (3.5), derivando-a duas vezes em relação a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x} , obtendo as seguintes formas diferencial e integral

As Equações em (3.12) possuem, conforme Equações (3.7) e (3.8), para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (x,y,t) \in }

SEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (i,j,k)}

, os seguintes análogos numéricos:

O campo vetorial aproximado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\textrm{'''H'''}}_m^* = ((f_m^*)_{xx}-(S_m^*)_x, (g_m^*)_{xx},(q_m^*)_{xx})}

é constante, de modo que a avaliação da Eq.  (3.13) sobre o elemento de conservação descrito pela Fig. 1a retorna, para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3}

Para computar as variáveis de marcha Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{yy}]_{i,j}^k}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{xy}]_{i,j}^k}
procede-se de modo  análogo, obtendo-se após todos os cálculos e simplificações:

Aborda-se na seção a seguir o cálculo das derivadas de primeira ordem das variáveis dinâmicas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_m} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3} .

3.2.2.3 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{x}]_{i,j}^k e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{y}]_{i,j}^k 3.2.2.3 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q m) {x}] {i,j}^k e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q m) {y}] {i,j}^k

As variáveis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_x]_{i,j}^k}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_y]_{i,j}^k}
podem ser determinadas por meio de uma estratégia  análoga àquela apresentada na seção [[#3.2.2.2 Avaliação de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{xx}]_{i,j}^k

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{xy}]_{i,j}^k

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{yy}]_{i,j}^k

|3.2.2.2]] anterior, bastando, para isso, construir leis diferenciais de conservação (já nas formas aproximadas)

e integrais

As avaliações das equações em (3.18) fornecem as inclinações equacionadas por

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \xi = x,y}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m = 1,2,3}

, em que

O esquema formado pelo conjunto de Equações (3.10), (3.14)-(3.16) e (3.19) é interessante apenas em problemas com soluções de comportamento suave. Para problemas com descontinuidades ou formação de choque, é necessário modificar o cálculo das variáveis em (3.14)-(3.16) e (3.19). A proposta apresentada neste artigo é uma variação do procedimento adotado por [37,38] e consiste num processo iterativo.

Para prosseguir, considere as seguintes formas regressivas e progressivas:

tal que o operador Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta _{\xi }^{\pm }}

seja definido como

As aproximações para as derivadas serão definidas como

e a função Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W}

[26,24]:

Observe que neste caso, as Equações (3.24) e (3.25) representam médias ponderadas das diferenças progressivas e regressivas descritas em (3.22) e (3.23). No caso de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha = 0} , (3.24) e (3.25) tornam-se diferenças finitas centrais [23]. Adota-se neste trabalho Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =1} .

Importante notar, no entanto, que as Equações (3.24) e (3.25) dependem da variável Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{i,j}^k}

que, por sua vez, depende de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{xx}]_{i,j}^k}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{yy}]_{i,j}^k}

, conforme Eq. (3.10). Dito de outra forma, para obter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{i,j}^k}

é necessário obter antes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{xx}]_{i,j}^k}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [(q_m)_{yy}]_{i,j}^k}

, e vice-versa. Para sanar esta dificuldade, construímos o processo iterativo descrito no Algoritmo 1.

     Faça Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): l = 1

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{yy}]_{i,j}^k
com as Equações (3.14) e  (3.15), respectivamente;

com a  Eq. (3.10);

     Calcule Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_x]_{i,j}^k
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_y]_{i,j}^k
com a Eq. (3.24);

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [(q_m)_{yy}]_{i,j}^k
com  a Eq.(3.25);

com a  Eq. (3.10);

da  malha.

A estabilidade do esquema CE/SE satisfaz a condição de Courant-Friedrichs-Lewy (CFL) que, para as equações de Saint Venant unidimensional e bidimensional são as restrições [23]

respectivamente, onde o número de Courant satisfaz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0 < CFL \leq 1} . Maiores detalhes sobre estabilidade do método CE/SE pode ser encontrado em [39].

3.3 Exemplos Numéricos

Esta seção tem por objetivo avaliar e validar o esquema numérico desenvolvido. Para isso, alguns problemas teste são avaliados, como testes de acurácia em uma e duas dimensões, bem como problemas clássicos de ruptura de barragem unidimensionais (subseção 3.3.1) e bidimensionais (subseção 3.3.2).

3.3.1 Exemplos Unidimensionais

Exemplo 1: O objetivo deste exemplo teste é verificar experimentalmente a ordem de acurácia do esquema CE/SE. Considere o sistema hiperbólico linear unidimensional

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): c

é uma constante dada. Impondo as condições inicias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \phi (x,0) =  -c^{-1}\sin (2\pi x)
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u(x,0) = 0

, é possível obter a seguinte solução exata [40,41]:

As soluções são computadas com as Equações (3.10), (3.14)-(3.16) e (3.19). A Tabela 1 apresenta os erros calculados nas normas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): L^1

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): L^2

, nas simulações realizadas sobre o domínio computacional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -1 \leq x \leq 1 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): c = 1 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): CFL = 0.8

 até atingir o tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 1.2

s, sendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n

o número de células no espaço. O experimento verifica a terceira  ordem de acurácia do esquema para ambas as variáveis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \phi 

.

Exemplo 2: Considera-se agora um problema ideal de ruptura de barragem sobre um domínio molhado, isto é, a quebra de barragem é instantânea, o fundo é plano e não existe resistência ao escoamento. As condições iniciais para esta configuração seguem o clássico problema de Riemann

O domínio considerado é Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{m}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \leq x \leq 2000\mbox{m}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x_0 = 1000\mbox{m}

e a solução analítica  pode ser encontrada em [42] ou [43]. Utiliza-se, para fins de análise  de comportamento, a mesma relação de equações do Ex. 1 anterior, isto é, Equações (3.10),  (3.14)-(3.16) e (3.19). As Fig. 2 e  Fig. 2a demonstram o comportamento da solução numérica em relação à analítica, calculadas no  tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 52\mbox{s}

, com Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n+1 = 201

pontos, incremento espacial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Delta x = 10\mbox{m}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): CFL = 0.8

e  profundidades iniciais a montante e a jusante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_l = 10\mbox{m}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r = 5\mbox{m}

e velocidade Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u
para o problema de ruptura de  barragem, computada no tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 52\mbox{s}

 pontos, incremento espacial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Delta x = 10\mbox{m}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r/h_l = 0.5

Utilizando o Algoritmo 1, sob as mesmas condições e com os mesmos parâmetros, obtém-se os gráficos constantes na Fig. 3. Observa-se que esta solução numérica é superior à anterior, sobretudo no que tange às regiões de rápidas mudanças. Conforme Zhang et al. (2012) [23], a razão Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r/h_l

é um importante índice para  julgar a aplicabilidade e a acurácia de esquemas numéricos no modelo 1D de ruptura de barragem. Segundo os  mesmos autores, os regimes de escoamentos subcrítico e supercrítico existem simultaneamente num canal  sem fricção, horizontal e retangular, quando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r/h_l < 0.138

. Altera-se, neste sentido, estes parâmetros para uma razão Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r/h_l = 0.001

e velocidade Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u
para o problema de ruptura de  barragem, no tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 52\mbox{s}

pontos e respectivo incremento espacial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Delta x = 10\mbox{m}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r = 5\mbox{m}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): h_r = 0.01\mbox{m}

3.3.2 Exemplos Bidimensionais

Exemplo 3: Considere agora o sistema hiperbólico linear bidimensional

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): c

é uma constante dada. Para os dados iniciais Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \phi (x,y,0) =  -c^{-1}[\sin (2\pi x)+\sin (2\pi y)]
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u(x,y,0) = v(x,y,0) = 0

, este problema admite a seguinte solução exata [40,41]:

A Tabela 3 apresenta os erros numéricos calculados nas normas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): L^1

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): L^2

, no tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 0.2 s, com parâmetros especificados em Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): c = 1 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): CFL = 0.4

e domínio computacional definido em Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -1 \leq x, y  \leq 1

, sendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n_x \times n_y

o número de células utilizadas na discretização. O experimento confirma  novamente a acurácia de terceira ordem do esquema CE/SE.

Exemplo 4: Este problema hipotético bidimensional é um exemplo utilizado por [23,44,45]. Neste problema as velocidades iniciais são todas nulas, a profundidade a montante é de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 10} m, enquanto que a profundidade a jusante é assumida ser Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5} m ou Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.1} m. O domínio computacional consiste de uma região de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 200\mbox{m}\times 200\mbox{m}} , com uma parede que se estende paralelamente ao eixo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y} , tendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 10} m de largura e está centrada em Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x = 100} m. A falha é suposta ser instantânea, possui Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 75} m de extensão a partir de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y = 95} m. O canal é horizontal e desconsidera-se resistência ao escoamento. Espera-se a formação de uma frente de choque após o rompimento. Os resultados em Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t = 7.2} s são computados a partir de uma malha uniforme composta por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 101 \times 101}

células e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle CFL = 0.4}

. Constam nas Figuras 4 e fig_CVRB2D gráficos da profundidade, vetores velocidade e curvas de nível para o problema com profundidade inicial a jusante de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5} m, enquanto que as Figuras fig_RB2D1 e 4a referem-se ao problema com profundidade inicial a jusante de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.1} m. Os resultados são consistentes com aqueles presentes na literatura [23,44,45].

com Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): CFL = 0.4

Exemplo 5: Este problema teste visa avaliar a habilidade do esquema em preservar simetria. Considera-se um domínio de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 50\mbox{m}\times 50\mbox{m}}

com  condições iniciais:

No instante da falha da barragem, supõe-se que a parede circular seja removida completamente e, subsequentemente, formam-se ondas que se espalham radialmente. A solução numérica é computada numa malha retangular uniforme com Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 101\times 101}

células e o passo de tempo é tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle CFL = 0.6}

. A Fig. 5 apresenta o perfil da superfície da água Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.72} s após a hipotética falha na barragem circular. Vetores velocidade e curvas nível relativas à superfície Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h}

estão dispostos na Fig.  5a. A simetria da solução numérica é bem preservada e está de acordo com a  literatura [7,46].

(a), vetores velocidade e curvas de nível  (b), para a  solução calculada no tempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t = 0.72\mbox{s}
com Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): CFL = 0.6

3.4 Conclusões

Este artigo apresentou o desenvolvimento de um novo esquema CE/SE explícito para a solução das equações de águas rasas em uma e duas dimensões. Na formulação construída, as variáveis dinâmicas e, consequentemente, as leis diferencial e integral de conservação, foram aproximadas localmente por expansões de Taylor de segunda ordem. O conjunto de variáveis de marcha tornou-se constituído pelas variáveis fluxo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_m}

e suas  derivadas espaciais de primeira Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_x}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_y}

e segunda ordem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{xx}, (q_m)_{xy}}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (q_m)_{yy}}

. Os experimentos computacionais realizados indicaram acurácia de terceira ordem sobre os problemas hiperbólicos lineares testados. As soluções numéricas dos problemas teste de ruptura de barragem unidimensional e bidimensional, caracterizados pela formação de choque, descontinuidade ou simetria, apresentaram concordância com as soluções analíticas e/ou com a literatura. Conclui-se, com isso, que o esquema proposto possui considerável habilidade em capturar choques e descontinuidades fazendo com que seja uma boa ferramenta para análise de fluxo de ruptura de barragem.

Agradecimentos

Os autores agradecem à CAPES pelo apoio financeiro, ao Programa de Pós-Graduação em Métodos Numéricos em Engenharia (PPGMNE - UFPR) e ao Instituto Federal de Educação, Ciência e Tecnologia Catarinense, Campus Araquari, pelo apoio à pesquisa.

3.5 Referências

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