2015/02/05 by Antti Koskela, Koskela, Antti, Elias Jarlebring +1
Computer Science · Mathematics · #65F10 #65F60 #65L05 #65L20 #Differential Equations and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Modeling and Simulation Systems #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1502.01613
openalex publication_date 2015/02/05 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
Exponential integrators that use Krylov approximations of matrix functions\nhave turned out to be efficient for the time-integration of certain ordinary\ndifferential equations (ODEs). This holds in particular for linear homogeneous\nODEs, where the exponential integrator is equivalent to approximating the\nproduct of the matrix exponential and a vector. In this paper, we consider\nlinear inhomogeneous ODEs, y'(t)=Ay(t)+g(t), where the function g(t) is\nassumed to satisfy certain regularity conditions. We derive an algorithm for\nthis problem which is equivalent to approximating the product of the matrix\nexponential and a vector using Arnoldi's method. The construction is based on\nexpressing the function g(t) as a linear combination of given basis functions\n[\φi]i=0^\∞ with particular properties. The properties are such\nthat the inhomogeneous ODE can be restated as an infinite-dimensional linear\nhomogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE has\nproperties that directly allow us to extend a Krylov method for\nfinite-dimensional linear ODEs. Although the construction is based on an\ninfinite-dimensional operator, the algorithm can be carried out with operations\ninvolving matrices and vectors of finite size. This type of construction\nresembles in many ways the infinite Arnoldi method for nonlinear eigenvalue\nproblems. We prove convergence of the algorithm under certain natural\nconditions, and illustrate properties of the algorithm with examples stemming\nfrom the discretization of partial differential equations.\n