A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

Abstract

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.

Examining the Effect of Introducing a Link From Electrical Excitation to Calcium Dynamics in a Cardiomyocyte

“…We use Newton's Method as nonlinear solver, and at each Newton step we use BiCGSTAB as the linear solver. Complete details of the numerical method can be found in [11,19].…”

“…, and potassium n(x, t). Section 3.1 describes the calcium signaling portion of the model without the presence of electrical excitation or mechanical contraction that was originally introduced in [12,14], extended in [13], and its numerics discussed with n sc = 2 in [9,10,19,20]. Section 3.2 introduces the electrical excitation that is connected to the calcium signaling in both the feedforward and feedback directions represented by link 1 ○ and link 2 ○ in Figure 1.1.…”

Examining the Electrical Excitation, Calcium Signaling, and Mechanical Contraction Cycle in a Heart Cell

“…We use the Newton method as non-linear solver, and the linear system in each Newton step is solved by BiCGSTAB as the linear solver. Complete details of the numerical method can be found in [5,10], and [4] contains practical details for the implementation, specifically the analytical derivation of the Jacobian matrix that is required for optimal convergence of the Newton method. BiCGSTAB is a Krylov subspace method, which only requires matrixvector products with the system matrix, not the matrix itself.…”

Challenges and opportunities for the simulation of calcium waves on modern multi‐core and many‐core parallel computing platforms

“…The following provides details on each of the components of the model, the calcium signaling portion of the model, the electrical excitation that is connected to the calcium signaling in both the feedforward and feedback directions represented by link 1 and link 2 in Figure 1 (a), and the mechanical contraction component that is also connected to the calcium signaling in both the feedback and feedforward directions represented by links 3 and 4 in Figure 1 (a). Table 1 collects the variables of the model with their units as well as their initial values.…”

“…Specifically, the equations are of parabolic type with key properties of mass conservation and smoothing behavior over time. The finite volume method (FVM) is the most appropriate discretization for this problem, since it guarantees mass conservation at the discrete level and will allows the incorporation of an advection term 2 ; by contrast, the finite element conserves mass, but needs more complicated handling of advection.…”