arXiv subjects

Jochen Schuetz

Publications and source records attributed to Jochen Schuetz.

2 recordsLinked to original sources

arXiv2026

Justification and structure- and asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation

We study a hyperbolic approximation ("hyperbolization") of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.

math.NA↗

arXiv2021

Two-derivative deferred correction time discretization for the discontinuous Galerkin method

In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations.

math.NA↗