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A method of fractional steps for scalar conservation laws without the CFL condition


Authors: Helge Holden and Nils Henrik Risebro
Journal: Math. Comp. 60 (1993), 221-232
MSC: Primary 65M12; Secondary 35L65
MathSciNet review: 1153165
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Abstract: We present a numerical method for the n-dimensional initial value problem for the scalar conservation law $ u{({x_1}, \ldots ,{x_n},t)_t} + \sum _{i = 1}^n{f_i}{(u)_{{x_1}}} = 0, u({x_1}, \ldots ,{x_n},0) = {u_0}({x_1}, \ldots ,{x_n})$. Our method is based on the use of dimensional splitting and Dafermos's method to solve the one-dimensional equations. This method is unconditionally stable in the sense that the time step is not limited by the space discretization. Furthermore, we show that this method produces a subsequence which converges to the weak entropy solution as both the time and space discretization go to zero. Finally, two numerical examples are discussed.


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  • [1] Frode Bratvedt, Kyrre Bratvedt, Christian F. Buchholz et al., Front tracking for petroleum reservoirs, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992, pp. 409–427. MR 1190515
  • [2] Edward Conway and Joel Smoller, Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95–105. MR 0192161
  • [3] Michael Crandall and Andrew Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), no. 3, 285–314. MR 571291, 10.1007/BF01396704
  • [4] Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 0303068
  • [5] S. K. Godunov, Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271-306 (Russian).
  • [6] Helge Holden and Lars Holden, On scalar conservation laws in one dimension, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992, pp. 480–509. MR 1190518
  • [7] H. Holden, L. Holden, and R. Høegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl. 15 (1988), no. 6-8, 595–602. Hyperbolic partial differential equations. V. MR 953567, 10.1016/0898-1221(88)90282-9
  • [8] S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-243.
  • [9] N. Kuznetsov, Weak solution of the Cauchy problem for a multi-dimensional quasi-linear equation, Math. Notes 2 (1967), 733-739.
  • [10] A. I. Vol′pert, Spaces 𝐵𝑉 and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1153165-5
Keywords: Dimensional splitting, scalar conservation law, fractional steps, numerical methods
Article copyright: © Copyright 1993 American Mathematical Society