Related papers: Arbitrary high-order, conservative and positive pr…
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…
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…
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…
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…
In \cite{BDM2003} the modified Patankar-Euler and modified Patankar-Runge-Kutta schemes were introduced to solve positive and conservative systems of ordinary differential equations. These modifications of the forward Euler scheme and…
The Deferred Correction (DeC) is an iterative procedure, characterized by increasing accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of…
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…
In [1] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for…
Positivity preservation of key physical quantities in the context of fluid flows, such as density and internal energy, is an essential property of a numerical scheme as otherwise the solution lacks physical relevance and has a not…
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the…
In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent…
Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the…
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…
We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the…
The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE…
We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1,…
We propose a new paradigm for designing efficient p-adaptive arbitrary high order methods. We consider arbitrary high order iterative schemes that gain one order of accuracy at each iteration and we modify them in order to match the…
In this paper, we demonstrate that the explicit ADER approach as it is used inter alia in [1] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [2]. By using this fact, we are able to embed…
Spectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential…
Spectral deferred correction (SDC) methods are an attractive approach to iteratively computing collocation solutions to an ODE by performing so-called sweeps with a low-order time stepping method. SDC allows to easily construct high order…