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Exponential integrators that use Krylov approximations of matrix functionshave turned out to be efficient for the time-integration of certain ordinarydifferential equations (ODEs). This holds in particular for linear homogeneousODEs, where the exponential integrator is equivalent to approximating theproduct of the matrix exponential and a vector. In this paper, we considerlinear inhomogeneous ODEs, $y'(t)=Ay(t)+g(t)$, where the function $g(t)$ isassumed to satisfy certain regularity conditions. We derive an algorithm forthis problem which is equivalent to approximating the product of the matrixexponential and a vector using Arnoldi's method. The construction is based onexpressing the function $g(t)$ as a linear combination of given basis functions$[\phi_i]_{i=0}^\infty$ with particular properties. The properties are suchthat the inhomogeneous ODE can be restated as an infinite-dimensional linearhomogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE hasproperties that directly allow us to extend a Krylov method forfinite-dimensional linear ODEs. Although the construction is based on aninfinite-dimensional operator, the algorithm can be carried out with operationsinvolving matrices and vectors of finite size. This type of constructionresembles in many ways the infinite Arnoldi method for nonlinear eigenvalueproblems. We prove convergence of the algorithm under certain naturalconditions, and illustrate properties of the algorithm with examples stemmingfrom the discretization of partial differential equations.

The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs