Selected Topics in Approximate Solutions of Nonlinear Conservation Laws. High-resolution Central Schemes

Central schemes offer a simple and versatile approach for computing approximate solutions of non-linear systems of hyperbolic conservation laws and related PDEs. The solution of such problems often involves the spontaneous evolution of steep gradients. The multiscale aspect of these gradients poses a main computational challenge for their numerical solution. Central schemes utilize a minimal amount of information on the propagation speeds associated with the problems, in order to accurately detect these steep gradients. This information is then coupled with high-order, non-oscillatory reconstruction of the approximate solution in ‘the direction of smoothness’: that is, information of smoothness does not cross regions of steep gradients. The use of central stencils enables us to realize the reconstructed solutions through simple quadratures. In this manner, central schemes avoid the intricate and time-consuming details of the eigen-structure of the underlying PDEs, and in particular, the use of (approximate) Riemann solvers, dimensional splitting, etc. The resulting family of central schemes offers relatively simple, “black-box” solvers for a wide variety of problems governed by multi-dimensional systems of non-linear hyperbolic conservation laws and related convection-diffusion problems. We highlight several features of this new class of central schemes. Scalar equations. Both the secondand third-order schemes were shown to have variation bounds, which in turn yield convergence with precise error estimates, as well as entropy and (multidimensional) L-stability estimates. Systems of equations. Extension to systems is carried out by component-wise application of the scalar framework. It is in this context that our central schemes offer a remarkable advantage over the corresponding upwind framework. Multidimensional problems. Since we bypass the need for (approximate) Riemann solvers, multidimensional problems are solved without dimensional splitting. In fact, the class of central schemes is utilized for a variety of nonlinear transport equations. A partial list of more than 120 references can be found in CentPack, [4]. CentPack is a collection of freely distributed C++ routines that implement a number of high-order, non-oscillatory central schemes for hyperbolic systems of conservation laws in oneand two-space dimensions, ut+f(u)x+g(u)y = 0.