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35
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 54 (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 28 (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
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on ”Methods and Models of Kinetic Theory
, 2010
"... 2. Hyperbolic systems with stiff relaxations 3 3. Kinetic equations: the Euler regime 8 4. Linear transport equations: the diffusion regime 15 ..."
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Cited by 10 (5 self)
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2. Hyperbolic systems with stiff relaxations 3 3. Kinetic equations: the Euler regime 8 4. Linear transport equations: the diffusion regime 15
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.
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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Cited by 6 (0 self)
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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.
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 6 (2 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.
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 6 (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.
A SUBSONICWELLBALANCED RECONSTRUCTION SCHEME FOR SHALLOW WATER FLOWS
"... Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reco ..."
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Cited by 5 (3 self)
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Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reconstruction that allows to derive a subsonicwellbalanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semidiscrete entropy inequality. An application to the EulerPoisson system is proposed. 1.
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 5 (0 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.
On the Computation of Roll Waves
 Math. Model. Num. Anal
, 2000
"... incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the nume ..."
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Cited by 5 (2 self)
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incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the numerical approximation of this problem is its linear instability. Numerical roundoff error can easily overtake the numerical solution and yields false roll wave solution at the steady state.