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Mathematics of Computation

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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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Keywords: Dimensional splitting, scalar conservation law, fractional steps, numerical methods
Article copyright: © Copyright 1993 American Mathematical Society