NUMERICAL SIMULATION OF NON-CLASSICAL WAVE FRONTS IN CONSERVATION LAWS

Abstract. We consider weak solutions for systems of conservation laws of type ut + f1(u)x1 + · · ·+ f2(u)xd = 0 in D × (0,∞), D ⊆ R d. Here u = u(x, t) ∈ U ⊂ Rm is the unknown and f1,..., fd ∈ C1(U,Rm) are the given fluxes. Typical solutions of associated initial-boundary value problems involve shock waves. Here we are interested in situations where so-called non-classical shock waves appear. In contrast to the well-understood case of classical Laxian waves (e.g. hydromechanial shock waves in gas dynamics) non-classical waves are not fully determined by the characteristic information. In the talk we will first discuss applications with non-classical waves modeling crucial physi-cal effects. Examples cover liquid-vapour flow in compressible fluid mechanics, the dynamics of shape-memory alloys in solid mechanics, and the flow of immiscible incompressible fluids in porous media. Non-Laxian shock waves appear in these applications as fronts separating different phases or materials. In general non-classical waves cannot be approximated numerically by standard schemes for conservation laws since a fine tuning of the numerical entropy dissipation is necessary. We wil