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Wellbalanced finite volume evolution Galerkin methods
"... for the shallow water equations ..."
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Finite Volume Schemes For Multidimensional Hyperbolic Systems Based On The Use Of Bicharacteristics
"... In this survey paper we present an overview on recent results for the bicharacteristics based finite volume schemes, the socalled finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multidimensional hyperbolic conservation laws. They combine the usually conflicting ..."
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In this survey paper we present an overview on recent results for the bicharacteristics based finite volume schemes, the socalled finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multidimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteritics, or bicharacteritics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of multidimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to study longtime behaviour of the FVEG schemes. We present several numerical experiments which confirm the fact that the FVEG methods are wellsuited for longtime simulations. Key words Multidimensional finite volume methods, Bicharacteristics, Hyperbolic systems, Wave equation, Euler equations 1
On the Connection between some RiemannSolver Free Approaches to the Approximation of MultiDimensional Systems of Hyperbolic Conservation Laws
 J. Comput. Phys
"... Abstract. In this paper, we present some interesting connections between a number of Riemannsolver free approaches to the numerical solution of multidimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively ..."
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Abstract. In this paper, we present some interesting connections between a number of Riemannsolver free approaches to the numerical solution of multidimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp’s equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.
A THREEDIMENSIONAL, UNSPLIT GODUNOV METHOD FOR SCALAR CONSERVATION LAWS ∗
"... Abstract. Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transpo ..."
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Abstract. Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transport in Darcy models of saturated subsurface flow. An important characteristic of these problems is that the velocity field is not known analytically. Instead, an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization. In this paper, we present a customized threedimensional finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles. The method is an extension of an algorithm by Bell, Dawson, and Shubin (BDS), which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions. The original BDS algorithm is a variant of unsplit, higherorder Godunov methods based on construction of a limited bilinear profile within each computational cell. Here we present a threedimensional extension of the original BDS algorithm that is based on a limited trilinear profile within each cell. We compare this new method to several other unsplit approaches, including piecewise linear methods, piecewise parabolic methods, and wave propagation schemes. Key words. Godunov method, scalar conservation law, linear advection AMS subject classifications. 3504, 35L65 DOI. 10.1137/100809520
A genuinely multidimensional relaxation scheme for hyperbolic conservation laws
 Proceedings of the Seventh Asian CFD conference, Bangalore
, 2007
"... Abstract A new genuinely multidimensional relaxation scheme is proposed. Based on a new discrete velocity Boltzmann equation, which is an improvement over previously introduced relaxation systems in terms of isotropic coverage of the multidimensional domain by the foot of the characteristic, a fin ..."
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Abstract A new genuinely multidimensional relaxation scheme is proposed. Based on a new discrete velocity Boltzmann equation, which is an improvement over previously introduced relaxation systems in terms of isotropic coverage of the multidimensional domain by the foot of the characteristic, a finite volume method is developed in which the fluxes at the cell interfaces are evaluated in a genuinely multidimensional way, in contrast to the traditional dimensionbydimension treatment. This algorithm is tested on some benchmark test problems for hyperbolic conservation laws.
Impact Factor: 0.4 · DOI: 10.1007/s104920060012z · Source: OAI CITATIONS
"... Finite volume schemes for multidimensional hyperbolic systems based on the use of bicharacteristics ..."
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Finite volume schemes for multidimensional hyperbolic systems based on the use of bicharacteristics
Compact Finite Difference Investigation of Pressure Field Governed by a Three Dimensional Wave Equation
"... Numerical Simulation of three dimensional wave equation is conducted in a cubic domain which contains two parts, part A and part B. Part A of the domain is a Ishaped empty channel, containing air, and part B, containing a solid medium, is the difference between the whole cubic domain and part A. A ..."
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Numerical Simulation of three dimensional wave equation is conducted in a cubic domain which contains two parts, part A and part B. Part A of the domain is a Ishaped empty channel, containing air, and part B, containing a solid medium, is the difference between the whole cubic domain and part A. A Dirichlet type boundary condition is specified at the inlet boundary. At all other boundaries of the cubic domain a null Neumann boundary conditions are imposed. As an initial condition a null pressure distribution is specified everywhere in the cubic domain except at the inlet boundary. All spatial derivatives are calculated using a compact finite difference scheme. Computations are advanced in time using a compact third order RungeKutta scheme. These computations have to be performed twice in succession for each subtime step of the RungeKutta scheme. The numerical code is successfully tested against an exact solution. This issue is followed by the discussion of a threedimensional simulation. These results, for pressure, are then integrated over the entire surface of the end plane to introduce the load applied to the plane. Both results indicates that the compact finite difference simulation of the wave equation produce a satisfactory and reliable result.
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"... On stability of the evolution Galerkin schemes applied to a twodimensional wave equation system 1 ..."
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On stability of the evolution Galerkin schemes applied to a twodimensional wave equation system 1
1 ON THE COMPARISON OF EVOLUTION GALERKIN AND DISCONTINUOUS GALERKIN SCHEMES
"... www.tuharburg.de/math/hp/lukacova The aim of this paper is to compare some recent numerical schemes for solving hyperbolic conservation laws. We consider the flux vector splitting finite volume methods, finite volume evolution Galerkin scheme as well as the discontinuous Galerkin scheme. All scheme ..."
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www.tuharburg.de/math/hp/lukacova The aim of this paper is to compare some recent numerical schemes for solving hyperbolic conservation laws. We consider the flux vector splitting finite volume methods, finite volume evolution Galerkin scheme as well as the discontinuous Galerkin scheme. All schemes are constructed using time explicit discretization. We present results of numerical experiments for the shallow water equations for continuous as well as discontinuous solutions and compare accuracy and computational efficiency of the considered methods.
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"... Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying ..."
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Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying