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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.
Convergence Analysis For OperatorSplitting Methods Applied To Conservation Laws With Stiff Source Terms
, 1998
"... . We analyze the order of convergence for operator splitting methods applied to conservation laws with sti# source terms. We suppose that the source term q(u) is dissipative. It is proved that the L 1 erro ..."
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Cited by 5 (3 self)
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.<F3.818e+05> We analyze the order of convergence for operator splitting methods applied to conservation laws with sti# source terms. We suppose that the source term<F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05> q(u) is dissipative. It is proved that the<F3.69e+05> L<F2.742e+05> 1<F3.818e+05> error introduced by the time splitting can be bounded by<F3.69e+05><F3.818e+05><F3.69e+05><F3.714e+05><F3.69e+05><F3.818e+05><F3.69e+05> O(#t#q(u<F2.742e+05> 0<F3.818e+05><F3.714e+05> )#<F3.264e+05> L<F2.949e+05> 1<F3.818e+05> ), which is an improvement of the<F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05> O(Q#t) upper bound, where<F3.69e+05> #t<F3.818e+05> is the splitting time step,<F3.69e+05> Q<F3.818e+05> is the Lipschitz constant of<F3.69e+05><F3.818e+05> q, or<F3.69e+05> Q<F3.818e+05> =<F3.264e+05> maxu<F3.714e+05><F3.69e+05> q<F3.659e+05> #<F3.818e+05><F3.69e+05><F3.818e+05><F3.714e+05> (u)<F3.818e+05> in case<F3.69e+05> q<F3.818e+05> is smooth. A generic model w...
Convergence Analysis for Operator Splitting Methods to Conservation Laws with Stiff Source Terms
, 1996
"... We analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms. We suppose that the source term q(u) is dissipative. It is proved that the L 1 error introduced by the timesplitting can be bounded by O(\Deltatkq(u 0)k L 1), which is an improve ..."
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We analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms. We suppose that the source term q(u) is dissipative. It is proved that the L 1 error introduced by the timesplitting can be bounded by O(\Deltatkq(u 0)k L 1), which is an improvement of the O(Q\Deltat) upper bound, where \Deltat is the splitting time step, Q is the Lipschitz constant of q or Q = max u jq 0 (u)j in case q is smooth. A generic model with a special form of stiff source is also investigated. We propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term. Our results are derived by using parabolic regularizations, rather than using the Kuznetsov's approximation theory which has been employed as a standard approach for error analysis to the dimensional or timesplitting methods. Numerical examples are presented to illustrate the theoretical results.
Finite Volume Schemes for Nonhomogeneous Scalar Conservation Laws: Error Estimate
"... In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law u t + div F (x; t; u) = q(x; t; u) with initial condition u(x; 0) = u 0 (x). The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given b ..."
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In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law u t + div F (x; t; u) = q(x; t; u) with initial condition u(x; 0) = u 0 (x). The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semiimplicit in the stiff case) and the entropy solution. The order of these estimates is h^1/4 in spacetime L¹norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case.
Error estimates for the approximate solution of a nonlinear hyperbolic equation with source term given by Finite Volume Scheme
, 1997
"... In this paper, we study a finite volume approximation of a nonlinear hyperbolic equation with source term q where q is a C 1 function from IR 3 to IR, nonincreasing w.r.t. the third variable with a third derivative which is bounded. We also suppose that q(:; :; 0) j 0. We suppose that u0 is in B ..."
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In this paper, we study a finite volume approximation of a nonlinear hyperbolic equation with source term q where q is a C 1 function from IR 3 to IR, nonincreasing w.r.t. the third variable with a third derivative which is bounded. We also suppose that q(:; :; 0) j 0. We suppose that u0 is in BV (IR N ). We use an explicit scheme with an implicit discretization for the source term. The aim of this article is to give an error estimate between the finite volume approximate solution and the unique entropy solution of the equation. Dans cet article, on 'etudie la solution approch'ee par une m'ethode de type Volumes Finis d'une 'equation hyperbolique non lin'eaire avec un terme source q. q est une fonction C 1 de IR 3 dans IR, d'ecroissante par rapport `a la troisi`eme variable et de d'eriv'ee par rapport `a cette variable born'ee. On suppose de plus que q(:; :; 0) j 0. Par hypoth`ese u0 est dans BV (IR N ). Le sch'ema consid'er'e est explicite en temps avec une discr'etisation...
First and second order error estimates for the Upwind Source at Interface method *
"... Abstract The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced in [29] is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove Lperror estimates, 1< = p <+ ..."
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Abstract The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced in [29] is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove Lperror estimates, 1< = p <+1, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem [6]. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question to derive second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the U.S.I. method. 1 Introduction We consider the initial value problem for a transport equation with nonlinearsource term, in one space dimension, @tu + @xu = B(x, u), t 2 R+, x 2 R, (1.1) u(0, x) = u0(x) 2 Lp(R) " L1(R), 1 < = p < +1, (1.2)
Convergence of MUSCL Relaxing Schemes for Conservation Laws with Stiff Source Terms
"... We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l 1  and TV stable in the se ..."
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We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l 1  and TV stable in the sense that they are bounded by a constant independent of the relaxation parameter ffl, the Lipschitz constant of the stiff source term and the time step \Deltat. The Lipschitz constant of the l 1 continuity in time for the MUSCL relaxing schemes is shown to be independent of ffl and \Deltat. The convergence of the relaxing schemes to the corresponding MUSCL relaxed schemes is then established.
ON STABILITY ISSUES FOR IMEX SCHEMES APPLIED TO 1D SCALAR HYPERBOLIC EQUATIONS WITH STIFF REACTION TERMS
"... Abstract. The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, ImplicitExplicit RungeKutta (IMEXRK) schemes are particularly useful as high order time integrators ..."
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Abstract. The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, ImplicitExplicit RungeKutta (IMEXRK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons. Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEXRK schemes, and study the conditions that guarantee that [0, 1] is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes. 1.