Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods

Delis, A. I.; Katsaounis, Theodoros

doi:10.1016/j.apm.2004.11.001

Cited by 52 publications

(46 citation statements)

References 39 publications

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“…In this section we briefly present the class of relaxation models and schemes introduced in [19] for homogeneous conservation laws and later extended in [9,11,24] to the nonlinear shallow water equations with the geometrical source term present.…”

Section: The Relaxation Model and Numerical Schemementioning

confidence: 99%

“…This simplicity can be of great significance when one has to solve large scale engineering problems or has to calculate complicated Jacobian matrices that follow the use of complex fluxes (such as a sediment transport flux). The amount of computational and theoretical results for relaxation schemes found in the literature has grown since they were first introduced, see for example [1,2,6,[9][10][11]13,21,[24][25][26] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Relaxation approximation to bed-load sediment transport

Delis

Papoglou

2008

Journal of Computational and Applied Mathematics

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In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit-explicit Runge-Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.

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Section: The Relaxation Model and Numerical Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relaxation approximation to bed-load sediment transport

Delis

Papoglou

2008

Journal of Computational and Applied Mathematics

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“…The constant relaxation scheme was introduced by Jin and Xin (1995) and has been extensively studied (Aregba-Driollet and Natalini 1996;Chalabi 1999;LeVeque and Pelanti 2001;Tadmor and Tang 2001). Relaxation schemes have been tried on problems like equations of gas dynamics (Banda 2005), shallow water systems (Delis and Katsounis 2005), and weakly-hyperbolic, conservation-formulation of Hamilton-Jacobi equations (Jin and Xin 1998).…”

Section: Introductionmentioning

confidence: 99%

A variable relaxation scheme for multiphase, multicomponent flow

Krishnamurthy

Gerritsen

2007

Transp Porous Med

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We propose a variable relaxation scheme for the simulation of 1D, two-phase, multicomponent flow in porous media. For these strongly nonlinear systems, traditional high order upwind schemes are impractical: Riemann solutions are not directly available when the phase behavior is complex, and the systems are weakly hyperbolic at isolated points. Relaxation schemes avoid the dependency on the eigenstructure and nonlinear Riemann solvers by approximating the original system with a strongly hyperbolic linear system. We exploit the known information about the eigenvalues to construct first order and second order variable relaxation schemes with much reduced numerical diffusion as compared to the standard relaxation formulations. The proposed second order variable relaxation scheme is competitive in accuracy and efficiency with a third order component-wise ENO reconstruction, and performs at least as well as second order component-wise TVD schemes.

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“…The problem is similar to that presented in References [34,40,41]. This problem, although it does not fit in the long wave framework, can serve as a good numerical test in terms of the wetting/drying process and conservation.…”

Section: Dam-break In a Channel With Topography And Frictionmentioning

confidence: 89%

A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains

Delis

Καζολέα

Kampanis

2007

Numerical Methods in Fluids

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SUMMARYThe propagation, runup and rundown of long surface waves are numerically investigated, initially in one dimension, using a well-balanced high-resolution finite volume scheme. A conservative form of the nonlinear shallow water equations with source terms is solved numerically using a high-resolution Godunov-type explicit scheme coupled with Roe's approximate Riemann solver. The scheme is also extended to handle two-dimensional complex domains. The numerical difficulties related to the presence of the topography source terms in the model equations along with the appearance of the wet/dry fronts are properly treated and extended. The resulting numerical model accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuities. Numerical results show very good agreement with previously presented analytical or asymptotic solutions as well as with experimental benchmark data.

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Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods

Cited by 52 publications

References 39 publications

Relaxation approximation to bed-load sediment transport

Relaxation approximation to bed-load sediment transport

A variable relaxation scheme for multiphase, multicomponent flow

A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains

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