Exponential integrators

2010/05/01 by Marlis Hochbruck, Alexander Ostermann · 2 citations
Mathematics · Engineering · #Numerical methods for differential equations #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #Exponential integrator #Exponential function #Integrator #Eigenvalues and eigenvectors #Discretization #Jacobian matrix and determinant #Mathematics #Applied mathematics #Matrix exponential #Function (biology) #Differential equation #Computer science #Mathematical analysis #Ordinary differential equation #Differential algebraic equation

paper · doi:10.1017/s0962492910000048

openalex publication_date 2010/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

Citations

• Exponential Integrators for Quantum-Classical Molecular Dynamics
• B-convergence results for linearly implicit one step methods
• Numerical integration of ordinary differential equations based on trigonometric polynomials
• Talbot quadratures and rational approximations
• Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants
• Reversible multiple time scale molecular dynamics
• On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation
• The Magnus expansion and some of its applications
• Fourth-Order Time-Stepping for Stiff PDEs
• On the exponential solution of differential equations for a linear operator
• Exponential Time Differencing for Stiff Systems
• Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems
• A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs
• Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
• Fast and Oblivious Convolution Quadrature

Cited by

• Implicit-explicit and split-explicit super-time-stepping methods
• Numerical methods for ordinary differential equations [wikipedia]

Related

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• Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs