Exponentially Convergent Algorithms for the Operator Exponential with Applications to Inhomogeneous Problems in Banach Spaces

Abstract

New exponentially convergent algorithms for the operator exponential generated by a strongly positive operator $A$ in a Banach space $X$ are proposed. These algorithms are based on representations by a Dunford--Cauchy integral along paths enveloping the spectrum of $A$ combined with a proper quadrature involving a short sum of resolvents where the choice of the integration path dramatically affects desired features of the algorithms. A parabola and a hyperbola are analyzed as the integration paths, and scales of estimates of dependence on the smoothness of initial data, i.e., of the initial vector and of the inhomogeneous right-hand side, are obtained. One of the algorithms possesses an exponential convergence rate for the operator exponential $e^{-At}$ for all $t\ge 0$ including the initial point. This allows one to construct an exponentially convergent algorithm for inhomogeneous initial value problems. The algorithm is parallelizable. It turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point. The efficiency of the proposed method is demonstrated by numerical examples.