Summary
New implicit schemes for solving a system of conservation laws in one space dimension are obtained by using the cubic-spline technique. By making use of certain perturbation terms, these implicit schemes have been transformed to dissipative schemes. The nonlinear instabilities appearing in the solution in the narrow shock region have been damped by applying the automatic switched Shuman-filter method. Four test examples with continuous and discontinuous initial conditions have been solved to illustrate the theory. The proposed method has been extended to solve a system of conservation laws in two space dimensions.
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Holla, D.N., Jain, P.C. Implicit dissipative schemes for solving systems of conservation laws. J Eng Math 13, 257–270 (1979). https://doi.org/10.1007/BF00036674
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DOI: https://doi.org/10.1007/BF00036674