Philipp Öffner
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Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations Open
http://dx.doi.org/10.13039/501100001659 Deutsche Forschungsgemeinschaft
Discontinuous Galerkin methods for the complete stochastic Euler equations Open
In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler equ…
Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations Open
We show that finite element discretizations of incompressible flow problems can be designed to ensure preservation/dissipation of kinetic energy not only globally but also locally. In the context of equal-order (piecewise-linear) interpola…
On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws Open
We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi :10 .1016 /j .jcp .2017 .01 .042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-…
Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods Open
High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often i…
Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty Open
In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos…
An optimization-based construction procedure for function space based summation-by-parts operators on arbitrary grids Open
We introduce a novel construction procedure for one-dimensional summation-by-parts (SBP) operators. Existing construction procedures for FSBP operators of the form $D = P^{-1} Q$ proceed as follows: Given a boundary operator $B$, the norm …
Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations Open
In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary…
Fully well balanced entropy controlled DGSEM for shallow water flows: global flux quadrature and cell entropy correction Open
In this paper we propose a high order DGSEM formulation for balance laws which embeds a general well balanced criterion agnostic of the exact steady state. The construction proposed exploits the idea of a global flux formulation to infer a…
Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form Open
Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known.…
On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws Open
We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like in…
Applications of Limiters, Neural Networks and Polynomial Annihilation in Higher-Order FD/FV Schemes Open
The construction of high-order structure-preserving numerical schemes to solve hyperbolic conservation laws has attracted a lot of attention in the last decades and various different ansatzes exist. In this paper, we compare several comple…
Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems Open
We investigate the consistency and convergence of flux-corrected finite element approximations in the context of nonlinear hyperbolic conservation laws. In particular, we focus on a monolithic convex limiting approach and prove a Lax--Wend…
Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction Open
Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that p…
Summation-by-parts operators for general function spaces: The second derivative Open
Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for s…
A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods Open
Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production–destruction system (PDS) irrespective of the chosen time step size. Although they are now o…
Summation-by-Parts Operators for General Function Spaces Open
Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that th…
Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction Open
Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that p…
Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction Open
We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit simi…
A necessary condition for non oscillatory and positivity preserving time-integration schemes Open
Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not y…