A high order discretization technique for singularly perturbed differential equations

The compressible Navier-Stokes equations converge toward their incompressible counterpart as the Mach number ε approaches 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 impose special requirements on numerical methods, causing standard discretization techniques to often fail in efficiently computing an accurate approximation. One solution is to split the equations into a stiff and a non-stiff part and then solve the stiff part implicitly and the non-stiff part explicitly in time. This procedure results in an IMEX method, with the crucial aspect being the choice of the splitting. In this thesis, the novel RS-IMEX splitting—which uses the ε → 0 limit to split the equations via linearization—is coupled with high-order IMEX Runge-Kutta schemes. The resulting method is applied to various singularly perturbed differential equations, and its behavior for ε << 1 is investigated. This is done in the following steps: First, the method is applied to a class of ordinary differential equations, and it is proven how the resulting discretization suffers from order reduction. To this end, it is shown that the convergence behavior depends on ε and that order reduction depends primarily on the implicit part of the discretization. This leads to improved convergence behavior compared to an established splitting method. Numerical computations demonstrate the influence of order reduction, and a comparison with standard methods is provided. Second, the isentropic Euler equations are considered to investigate the resulting method in the context of a weakly compressible flow. A discontinuous Galerkin method is used for the spatial discretization. It is proven that the resulting method is consistent in the limit as ε approaches 0, i.e., the overall algorithm is asymptotically consistent. Subsequently, numerical computations are used to investigate stability and accuracy. Overall, the method proposed in this thesis is a high-order discretization for singularly perturbed differential equations that is consistent in the ε → 0 limit and exhibits the desired behavior in the low-Mach setting.

DOI 10.18154/RWTH-2018-228660