Related papers: Flux-Balanced Patankar-type Schemes for the Compre…
A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions…
Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production-destruction systems. They adapt explicit Runge-Kutta schemes to ensure positivity and conservation irrespective of…
In this study, a family of second order process based modified Patankar Runge-Kutta schemes is proposed with both the mass and mole maintained in balance while preserving the positivity of density and pressure with the time step determined…
Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there are only little results on their stability or robustness. We suggest two approaches to analyze the…
We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative…
Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time…
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are…
Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and…
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 of…
In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the…
Many natural processes, such as chemical reactions and wave dynamics, are modeled as production-destruction (PD) systems that obey positivity and linear conservation laws. Classical time integrators do not guarantee positivity and can…
This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the…
In this paper, we perform stability analysis for a class of second and third order accurate strong-stability-preserving modified Patankar Runge-Kutta (SSPMPRK) schemes, which were introduced in [4,5] and can be used to solve convection…
In this work modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity preserving for any time…
We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density…
We develop structure-preserving numerical methods for the compressible Euler equations, employing potential temperature as a prognostic variable. We construct three numerical fluxes designed to ensure the conservation of entropy and total…
Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do…
Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form $f^m_\jph =…
Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure…
This paper presents high-order Runge-Kutta (RK) discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to…