In [BDM03] 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 Heun's method guarantee positivity and conservation irrespective of the chosen time step size. In this paper we introduce a general definition of modified Patankar-Runge-Kutta schemes and derive necessary and sufficient conditions to obtain first and second order methods. We also introduce two novel families of second order modified Patankar-Runge-Kutta schemes.
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 in a way to ensure positivity and conservation irrespective of the time step size.The first two members of this class, the first order MPE scheme and the second order MPRK22(1) scheme, were introduced in [BDM03] and have been successfully applied in a large number of applications. Recently, we introduced a general definition of MPRK schemes and presented a thorough investigation of first and second order MPRK schemes in [KM17].A potentially third order Patankar-type method was introduced in [FS11]. This method uses the MPRK22(1) scheme of [BDM03] as a predictor and a modification of the BDF(3) multistep method as a corrector. It restricts to the MPRK22(1) approximation, whenever the positivity of the corrector cannot be guaranteed. Hence, this method is at most third order accurate and at least second order accurate.In this paper we continue the work of [KM17] and present necessary and sufficient conditions for third order MPRK schemes. For the first time, we introduce MPRK schemes, which are third order accurate independent of the specific positive and conservative system under consideration. The theoretical results derived within the first part are subsequently confirmed by numerical experiments for the entire domain of linear and nonlinear as well as nonstiff and stiff systems of differential equations.
We present a parallel matrix-free implicit finite volume scheme for the solution of unsteady three-dimensional advection-diffusion-reaction equations with smooth and Dirac-Delta source terms. The scheme is formally second order in space and a Newton-Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix-vector product required is hardcoded without any approximations, obtaining a matrix-free method that needs little storage and is well suited for parallel implementation. We describe the matrix-free implementation of the method in detail and give numerical evidence of its second order convergence in the presence of smooth source terms. For non-smooth source terms the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long time simulation of calcium flow in heart cells and show its parallel scaling.Keywords: Finite volume method; Dirac delta distribution; Matrix-free Newton-Krylov method; Calcium waves; Parallel computing I INTRODUCTION Advection-diffusion-reaction systems occur in a wide variety of applications, as for instance heat transfer or transport-chemistry problems. Long-time simulations of such real-life applications often require implicit methods on very fine computational grids, to resolve the desired accuracy, especially in three dimensions. When performing such simulations with methods storing system matrices, the availability of memory becomes an issue already on relatively coarse meshes. One way to address this problem is the use of parallel methods, which distribute the workload among several CPUs and use the memory associated with these CPUs to solve larger systems. If this approach does not provide enough memory to obtain the desired resolution, parallel matrix-free methods are an excellent choice, since in general most of the memory is used to store system matrices. In addition, matrix-free methods require less communication compared to classical schemes, thus these are more suited for use on parallel architectures and allow for better scalability.To solve linear systems most matrix-free methods use Krylov subspace methods, which only require the results of matrix-vector products in every iteration. If these can be provided without storing the matrix, this leads to significant savings of memory and computations on high resolution meshes become feasible. A prominent example are Jacobian-free Newton Krylov (JFNK) methods, which approximate the Jacobian matrix by finite differences via function evaluations [1]. In the following we present another type of matrix-free Newton-Krylov method, which provides the matrix-vector products with the exact Jacobian by hardcoding the product. This method is thus specific to a given class of equations and grids and particularly suitable for computations on structured grids.
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