Computing semigroups with error control
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We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator A, a time t>0, an arbitrary initial vector u_0 and an error tolerance ϵ>0, the algorithm computes exp(tA)u_0 with error bounded by ϵ. The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain L^2(ℝ^d) that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like exp(-cN/log(N)) for N quadrature points, that remains stable as N→∞, and which is also suitable for infinite-dimensional operators. Numerical examples are given, including: Schrödinger and wave equations on the aperiodic Ammann–Beenker tiling, complex perturbed fractional diffusion equations on L^2(ℝ), and damped Euler–Bernoulli beam equations.
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