A tensor-based exponential integrator for diffusion--reaction equations in common curvilinear coordinates

TL;DR

This paper introduces a tensor-based exponential integrator for efficiently solving diffusion-reaction equations in common curvilinear coordinates, enabling large-scale simulations of pattern formation.

Contribution

It presents a novel split exponential Euler method combined with tensor techniques for structured discretizations, improving computational efficiency for complex geometries.

Findings

Successfully applied to large 2D and 3D problems with up to one million degrees of freedom.

Achieved stable and efficient time integration without severe step size restrictions.

Demonstrated capability to simulate Turing patterns in physically relevant systems.

Abstract

In this paper, we study a tensor-based method for the numerical solution of a class of diffusion--reaction equations defined on spatial domains that admit common curvilinear coordinate representations. Typical examples in 2D include disks (polar coordinates), and in 3D balls or cylinders (spherical or cylindrical coordinates) as well as spheres for problems involving the Laplace--Beltrami operator. The proposed approach is based on a carefully chosen finite difference discretization of the Laplace operators that yields matrices with a structured representation as sums of Kronecker products. For the time integration, we introduce a novel split variant of the exponential Euler method that effectively handles the stiffness and avoids the severe time step size restriction of classical explicit methods. By exploiting the peculiar form of the obtained discretized operators and the chosen splitting strategy, we compute the needed action of the $\varphi_1$ matrix function through suitable tensor-matrix products in a $\mu$-mode framework. We demonstrate the efficiency the approach on a wide range of physically relevant 2D and 3D examples of coupled diffusion--reaction systems generating Turing patterns with up to $10^6$ degrees of freedom.