On the implementation of exponential methods for semilinear parabolic equations
It addresses the computational bottleneck of exponential integrators for time-dependent PDEs, offering a more efficient implementation.
The paper proposes a new algorithm for implementing exponential methods for semilinear parabolic equations, achieving quadrature convergence rates of O(e^{-cK/ln K}) for operators and O(e^{-cK}) for scalar mappings.
The time integration of semilinear parabolic problems by exponential methods of different kinds is considered. A new algorithm for the implementation of these methods is proposed. The algorithm evaluates the operators required by the exponential methods by means of a quadrature formula that converges like $O(e^{-cK/\ln K})$, with $K$ the number of quadrature nodes. The algorithm allows also the evaluation of the associated scalar mappings and in this case the quadrature converges like $O(e^{-cK})$. The technique is based on the numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm.