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Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 78 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
A wavepropagation method for conservation laws and balance laws with spatially varying flux functions
 SIAM J. Sci. Comput
, 2002
"... Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a ge ..."
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Cited by 42 (5 self)
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Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A highresolution wavepropagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)− fi−1(Qi−1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be secondorder accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f(q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasisteady problems close to steady state. Key words. finitevolume methods, highresolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X
A Qscheme for a class of systems of coupled conservation laws with source term. Application to a twolayer 1D shallow water system
, 2001
"... Abstract. The goal of this paper is to construct a rstorder upwind scheme for solving the system of partial dierential equations governing the onedimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermudez and ..."
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Cited by 23 (6 self)
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Abstract. The goal of this paper is to construct a rstorder upwind scheme for solving the system of partial dierential equations governing the onedimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermudez and VazquezCendon [3, 26, 27] for solving onelayer shallow water equations, consisting in a Qscheme with a suitable treatment of the source terms. The diculty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to dene a suitable numerical scheme with global upwinding, we rst consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe’s method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Qscheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers ’ equations is considered. Then, the Qscheme obtained is applied to the twolayer shallow water system. Mathematics Subject Classication. 65M99, 76B55, 76B70.
High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallowwater systems
 Math. Comp
"... Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO r ..."
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Cited by 11 (3 self)
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Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the wellbalanced properties of the resulting schemes. Finally, we will focus on applications to shallowwater systems. 1.
The surface gradient method for the treatment of source terms in the shallowwater equations
 Journal of Computational Physics
, 2001
"... A novel scheme has been developed for data reconstruction within a Godunovtype method for solving the shallowwater equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstru ..."
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Cited by 10 (0 self)
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A novel scheme has been developed for data reconstruction within a Godunovtype method for solving the shallowwater equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstruction. This provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a Riemann solver. The main advantages are: (1) a simple centered discretization is used for the source terms; (2) the scheme is no more complicated than the conventional method for the homogeneous terms; (3) small perturbations in the water surface elevation can be accurately predicted; and (4) the method is generally suitable for both steady and unsteady shallowwater problems. The accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems. Excellent agreement has been obtained between the numerical predictions and analytical solutions. The results indicate that the new scheme is accurate, simple, efficient, and robust. c ° 2001 Academic Press Key Words: source terms; shallowwater equations; data reconstruction; highresolution method; Godunov method; MUSCL scheme.
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review
, 2012
"... Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schem ..."
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Cited by 10 (5 self)
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Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schemes that are efficient in these asymptotic regimes. The designing principle of AP schemes is to preserve, at the discrete level, the asymptotic limit that drives one (usually the microscopic) equation to its asymptotic (macroscopic) equation. An AP scheme is based on solving the microscopic equation, instead of using a multiphysics approach that couples different physical laws at different scales. When the small scale is not numerically resolved, an AP scheme automatically becomes a macroscopic solver for the limiting equation. The AP methodology offers simple, robust and efficient computational methods for a large class of multiscale kinetic, hyperbolic and other physical problems. This
Flux Difference Splitting and the Balancing of Source Terms and Flux Gradients.
, 1999
"... Flux difference splitting methods are widely used for the numerical approximation of homogeneous conservation laws where the flux depends only on the conservative variables. However, in many practical situations this is not the case. Not only are source terms commonly part of the mathematical model, ..."
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Cited by 9 (1 self)
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Flux difference splitting methods are widely used for the numerical approximation of homogeneous conservation laws where the flux depends only on the conservative variables. However, in many practical situations this is not the case. Not only are source terms commonly part of the mathematical model, but the flux can vary spatially even when the conservative variables do not. It is the discretisation of the additional terms arising from these two situations which is addressed in this work, given that a specific flux difference splitting method has been used to approximate the underlying conservation law. The discretisation is constructed in a manner which retains an exact balance between the flux gradients and the source terms when this is appropriate. The effectiveness of these new techniques, in both one and two dimensions, is illustrated using the shallow water equations, in which the additional terms arise from the modelling of bed slope and, in one dimension, breadth variation. Roe's scheme is chosen for the approximation of the conservation laws and appropriate discrete forms are constructed for the additional terms, not only in the first order case (which has been done before) but also in the presence of flux limited and slope limited high resolution corrections. The method is then extended to twodimensional ow where it can be applied on both quadrilateral and triangular grids.
An asymptotic high order masspreserving scheme for a hyperbolic model of chemotaxis
 SIAM J. Num. Anal
"... Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutio ..."
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Cited by 8 (2 self)
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Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.
Relaxation Schemes for the Shallow Water Equations
 INT. J. NUMER. METH. FLUIDS
, 2003
"... We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKut ..."
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Cited by 8 (0 self)
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We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present.
Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms
 Math. Comput
, 1999
"... Abstract. We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semilinear hyperbolic system with a second stiff sour ..."
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Abstract. We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semilinear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods. 1.