Abstract
Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu 1+f(u)x=0. We choose a class of distribution functionsG=G(t, x, ξ), which are compactly supported with respect to the artificial velocityξ. This can be done in an optimal way, i.e. so that theξ-integral of the solution of the linear kinetic equationG t+ξGx=0 solves the nonlinear conservation law exactly.
Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.
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Bäcker, M., Dressler, K. A kinetic method for strictly nonlinear conservation laws. Z. angew. Math. Phys. 42, 243–256 (1991). https://doi.org/10.1007/BF00945796
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DOI: https://doi.org/10.1007/BF00945796