[Paper Review] Local Exponential Methods: a domain decomposition approach to exponential time integration of PDEs

Luca Bonaventura|arXiv (Cornell University)|May 9, 2015

Numerical methods for differential equations19 references3 citations

TL;DR

This paper proposes Local Exponential Methods (LEM), a domain decomposition approach that replaces global matrix exponentials in exponential time integration with localized, parallelizable computations over overlapping subdomains. By leveraging spatial decay in matrix exponentials—supported by Iserles' theoretical bounds—LEM reduces computational cost and enables storage of small local exponential matrices, achieving comparable accuracy to standard methods at higher Courant numbers with improved scalability.

ABSTRACT

A local approach to the time integration of PDEs by exponential methods is proposed, motivated by theoretical estimates by A.Iserles on the decay of off-diagonal terms in the exponentials of sparse matrices. An overlapping domain decomposition technique is outlined, that allows to replace the computation of a global exponential matrix by a number of independent and easily parallelizable local problems. Advantages and potential problems of the proposed technique are discussed. Numerical experiments on simple, yet relevant model problems show that the resulting method allows to increase computational efficiency with respect to standard implementations of exponential methods.

Motivation & Objective

• To reduce the computational cost of exponential time integration for PDEs by replacing global matrix exponentials with local, parallelizable computations.
• To exploit spatial locality in matrix exponentials, as theoretically supported by Iserles' decay estimates for sparse matrices.
• To enable storage of small local exponential matrices via Padé approximation, avoiding iterative Krylov methods.
• To improve scalability and efficiency in high-order and anisotropic discretizations, especially in atmospheric and oceanic modeling.
• To provide a robust, local alternative to directional splitting for problems with high vertical Courant numbers.

Proposed method

• Apply overlapping domain decomposition to partition the computational domain into subdomains, each with a buffer zone to minimize far-field influence.
• Compute local matrix exponentials for each subdomain using Padé approximation with scaling and squaring, enabling storage and reuse.
• Use the exponential of the local matrix to advance the solution within each subdomain, treating interface nodes as shared.
• Ensure consistency by overlapping subdomains such that the influence of distant nodes is negligible, based on physical propagation speed and time step.
• Solve local problems independently and in parallel, reducing communication and improving scalability.
• Integrate with exponential Rosenbrock or other exponential time integrators to maintain high-order accuracy in time.

Experimental results

Research questions

• RQ1 Can local exponential matrices be used to approximate global matrix exponentials in PDE time integration while preserving accuracy?
• RQ2 Does domain decomposition with overlapping subdomains reduce computational cost compared to standard exponential methods?
• RQ3 To what extent does the buffer zone size affect efficiency, especially at high Courant numbers?
• RQ4 Can local exponential matrices be stored and computed efficiently via Padé approximation in high-order or anisotropic discretizations?
• RQ5 Is LEM competitive with semi-implicit or semi-Lagrangian methods in models with strong anisotropy, such as atmospheric and oceanic simulations?

Key findings

• The Local Exponential Method (LEM) achieves second-order accuracy comparable to a fourth-order explicit Runge-Kutta method when run at a Courant number of 7 in a linear advection test case.
• For a nonlinear two-dimensional Burgers equation on anisotropic meshes, LEM maintained accuracy at a vertical Courant number of 6 and horizontal Courant number below 1, using column-wise domain decomposition.
• The method reduces computational cost compared to standard exponential methods, despite overhead from overlapping buffer regions, due to efficient local matrix computation and storage.
• Storing local exponential matrices via Padé approximation avoids the need for iterative Krylov methods, significantly reducing per-step cost when subdomain size is small.
• LEM shows promise for high-order discontinuous Galerkin methods and anisotropic meshes, where high Courant numbers are common and traditional methods become inefficient.
• The approach provides a fully local alternative to directional splitting in models with strong vertical anisotropy, such as numerical weather prediction and ocean modeling.