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Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
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A Wave Propagation Method for Three Dimensional Hyperbolic Problems
, 1996
"... A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. ..."
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Cited by 29 (6 self)
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A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model crossderivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.
Highresolution FEMFCT schemes for multidimensional conservation laws
"... The fluxcorrectedtransport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory loworder method of upwind type is derived by elimination of negative offdiagonal entries of the discrete transport ope ..."
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Cited by 28 (16 self)
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The fluxcorrectedtransport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory loworder method of upwind type is derived by elimination of negative offdiagonal entries of the discrete transport operator. The difference between the discretizations of high and low order is decomposed into a sum of skewsymmetric antidiffusive fluxes. An iterative flux limiter independent of the time step is proposed for implicit schemes. The nonlinear antidiffusion is incorporated into the solution in the framework of a defect correction scheme preconditioned by the monotone loworder operator. In the case of a hyperbolic system, the global Jacobian matrix is assembled edgebyedge without resorting to numerical integration. Its loworder counterpart is constructed by rendering all offdiagonal blocks positive definite or adding scalar artificial diffusion proportional to the spectral radius of the Roe matrix. The coupled equations are solved in a segregated manner within an outer defect correction loop equipped with a blockdiagonal preconditioner. After a suitable synchronization, the correction factors evaluated for an arbitrary set of indicator variables are applied to the antidiffusive fluxes which are inserted into the global defect vector. The performance of the new algorithm is illustrated by numerical examples for scalar transport problems and the compressible Euler equations. Key Words: convectiondominated flows; hyperbolic conservation laws; flux correction; finite elements; implicit timestepping 1
Algebraic flux correction II. Compressible Euler equations
 FluxCorrected Transport: Principles, Algorithms, and Applications
, 2005
"... Summary. Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edgebyedge matrix assembly. A ..."
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Cited by 17 (12 self)
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Summary. Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edgebyedge matrix assembly. A generalization of Roe’s approximate Riemann solver is derived by rendering all offdiagonal matrix blocks positive semidefinite. Another usable loworder method is constructed by adding scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. The limiting of antidiffusive fluxes is performed using a transformation to the characteristic variables or a suitable synchronization of correction factors for the conservative ones. The outer defect correction loop is equipped with a blockdiagonal preconditioner so as to decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a strongly coupled solution strategy (global BiCGSTAB method with a blockGaußSeidel preconditioner) is introduced for applications which call for the use of large time steps. Various algorithmic aspects including the implementation of characteristic boundary conditions are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers. 1
Threedimensional Euler Computations using CLAWPACK
 in Conf. on Numer. Meth. for Euler and NavierStokes Eq
, 1995
"... this paper will be submitted for publication elsewhere. ..."
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Cited by 4 (2 self)
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this paper will be submitted for publication elsewhere.
Algebraic flux correction II. Compressible Flow Problems
"... Abstract Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCTlike algebraic flux correction schemes for the Euler equations of gas dynamics. In particular ..."
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Abstract Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCTlike algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCTconstrained L2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semiimplicit timestepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and nondifferentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudotime step is increased. 1
BLOCKSTRUCTURED ADAPTIVE MESH REFINEMENT THEORY, IMPLEMENTATION AND APPLICATION
"... Abstract. Structured adaptive mesh refinement (SAMR) techniques can enable cuttingedge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for ..."
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Abstract. Structured adaptive mesh refinement (SAMR) techniques can enable cuttingedge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for distributed memory computers. An overview of the background of commonly used finite volume discretizations for gas dynamics is included and typical benchmarks to quantify accuracy and performance of the dynamically adaptive code are discussed. Largescale simulations of shockinduced realistic combustion in nonCartesian geometry and shockdriven fluidstructure interaction with fully coupled dynamic boundary motion demonstrate the applicability of the discussed techniques for complex
Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods
"... In this paper, we present a collection of algorithmic tools for constraining highorder discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques lea ..."
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In this paper, we present a collection of algorithmic tools for constraining highorder discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques leads to an unconditionally stable implicit algorithm for steadystate computations. The implicit global problem for the mean values (coarse scales) has the computational structure of a finite volume method. The constrained derivatives (fine scales) are obtained by solving small local problems. The interscale transfer operators provide a twoway iterative coupling between the solutions to the global and local problems. Another highlight of this paper is a new approach to compatible gradient limiting for the Euler equations of gas dynamics. After limiting the conserved quantities, the gradients of the velocity and energy density are constrained in a consistent manner. Numerical studies confirm the accuracy and robustness of the proposed algorithms. Key words: hyperbolic conservation laws, discontinuous Galerkin methods, variational multiscale methods, slope limiting, constrained optimization 1.
the compressible
"... fluxcorrected transport algorithm for finite element simulation of ..."
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NUMERICAL METHODS FOR RIVER FLOW MODELLING ∗
"... In this paper we propose a new numerical scheme to simulate the river flow in the presence of a variable bottom surface. We use the finite volume methods, our approach is based on the technique described by D. L. George for shallow water equations [1]. The main goal is to construct the scheme, which ..."
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In this paper we propose a new numerical scheme to simulate the river flow in the presence of a variable bottom surface. We use the finite volume methods, our approach is based on the technique described by D. L. George for shallow water equations [1]. The main goal is to construct the scheme, which is well balanced, i.e. maintains not only some special steady states but all steady states which can occur. Furthermore this should preserve non–negativity of some quantities, which are essentially non–negative from their physical fundamental, for example cross section or depth. Our scheme can be extended to the second order accuracy. We also describe connections between the central and centralupwind schemes and the approximate Riemann solvers. 1.