Invariant-region-preserving discontinuous Galerkin methods for systems of hyperbolic conservation laws
This thesis is aimed at developing high order invariant-region-preserving (IRP) discontinuous Galerkin (DG) schemes solving hyperbolic conservation law systems. In particular, our focus is on designing an IRP limiter to modify the numerical solution and identifying sufficient conditions for IRP DG schemes by extending Zhang and Shu's work on positivity-preserving schemes for conservation laws [X. Zhang and C.-W. Shu., Journal of Computational Physics, 2010]. The one-dimensional model problems investigated are the p-system for two by two system and compressible Euler equations for three by three case. The feature that the invariant regions for these systems can be expressed by convex or concave functions of the solution to the system allows us to construct the IRP limiter in an explicit form, which therefore is easy for computer implementations. Rigorous analysis is presented to show that the IRP limiter does not destroy the order of approximation accuracy for smooth solutions, provided that the cell average of numerical solution is away from the boundary of the invariant region. For arbitrarily high order DG schemes solving hyperbolic conservation law systems, sufficient conditions are identified for cell averages to remain in the invariant region. We further extended these results to general multidimensional hyperbolic conservation law systems as long as (i) the system admits a global invariant region and (ii) the corresponding one-dimensional projected system shares the same invariant region. An application of these results has been investigated for two dimensional compressible Euler equations. Numerical tests on model problems have shown that the IRP limiter maintains the order of approximation for smooth solutions and helps damp oscillations near discontinuities. In addition, we designed second and third order IRP DG schemes for the viscous p-system by using the direct DG (DDG) diffusive flux proposed in [H. Liu and J. Yan., Communications in Computational Physics, 2010], considering the p-system and its viscous counterpart share the same invariant region. Numerical tests validate the desired properties of the IRP limiter for the viscous problem. The convergence of the viscous profiles to the entropy solution is illustrated from a numerical point of view.