ADAPTIVE SEMIDISCRETE CENTRAL-UPWIND SCHEMES FOR NONCONVEX HYPERBOLIC CONSERVATION LAWS

摘要

We discover that the choice of a piecewise polynomial reconstruction is crucial in computing solutions of nonconvex hyperbolic (systems of) conservation laws. Using semidiscrete central-upwind schemes, we illustrate that the obtained numerical approximations may fail to converge to the unique entropy solution or the convergence may be so slow that achieving a proper resolution would require the use of (almost) impractically fine meshes. For example, in the scalar case, all computed solutions seem to converge to solutions that are entropy solutions for some entropy pairs. However, in most applications, one is interested in capturing the unique (Kruzhkov) solution that satisfies the entropy condition for all convex entropies. We present a number of numerical examples that demonstrate the convergence of the solutions, computed with the dissipative second-order minmod reconstruction, to the unique entropy solution. At the same time, more compressive and/or higher-order reconstructions may fail to resolve composite waves, typically present in solutions of nonconvex conservation laws, and thus may fail to recover the Kruzhkov solution. In this paper, we propose a simple and computationally inexpensive adaptive strategy that allows us to simultaneously capture the unique entropy solution and to achieve a high resolution of the computed solution. We use the dissipative minmod reconstruction near the points where convexity changes and utilize a fifth-order weighted essentially nonoscillatory (WENO5) reconstruction in the rest of the computational domain. Our numerical examples (for one- and two-dimensional scalar and systems of conservation laws) demonstrate the robustness, reliability, and nonoscillatory nature of the proposed adaptive method.