A high order discretization technique for singularly perturbed differential equations
Kaiser, Klaus; Noelle, Sebastian (Thesis advisor); Schütz, Jochen (Thesis advisor); Munz, Claus-Dieter (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2018. - Dissertation, Hasselt University, 2018
The compressible Navier-Stokes equations converge towards their incompressible counterpart as the Mach number ε tends to zero. In the case of a weakly compressible flow, i.e. ε << 1, the resulting equations can be classified as singularly perturbed differential equations. Unfortunately, these equations set special requirements on numerical methods due to which standard discretization techniques often fail in efficiently computing an accurate approximation. One remedy is to split the equations into a stiff and a non-stiff part and then handle the stiff part implicitly and the non-stiff part explicitly in time. This procedure results in an IMEX method, with the crucial part being the choice of the splitting. In this thesis the novel RS-IMEX splitting, which uses the ε → 0 limit to split the equations by a linearization, is coupled with high order IMEX Runge-Kutta schemes. The resulting method is applied to different singularly perturbed differential equations and investigated in its behavior for ε << 1. This is done in the following steps: First, the method is applied to a class of ordinary differential equations and it is proven in which way the resulting discretization suffers from order reduction. For this, it is shown that the convergence behavior depends on ε and that order reduction mainly depends on the implicit part of the discretization. This leads to an improved convergence behavior compared to an established splitting. Numerical computations show the influence of order reduction and a comparison to standard methods is provided. Second, the isentropic Euler equations are considered to investigate the resulting method in the setting of a weakly compressible flow. For the spatial discretization a discontinuous Galerkin method is used. It is proven that the resulting method is consistent with the ε → 0 limit of the equations, i.e. the overall algorithm is asymptotically consistent. Then, with the help of numerical computations an investigation of stability and accuracy is provided. Overall, the method proposed in this thesis is a high order discretization for singularly perturbed differential equations which is consistent the ε → 0 limit and shows the desired behavior in the low Mach setting.