Abstract
A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u t + ∑ i=1 d V i(x, t)f i(u) x i = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.
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Lie, KA. A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients. BIT Numerical Mathematics 39, 683–700 (1999). https://doi.org/10.1023/A:1022339223716
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DOI: https://doi.org/10.1023/A:1022339223716