A High-order Exponential Integrator for Nonlinear Parabolic Equations with Nonsmooth Initial Data

Abstract

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential 𝑘-step method would have 𝑘 th -order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data.

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data

“…The development of efficient numerical methods that may have some uniform temporal convergence up to t = 0 is still challenging. In view of the low-regularity integrators recently developed for dispersive equations [26,35,37] and semilinear parabolic equations [33] this is possible and worth to be considered (at least for semi-discretization in time).…”

“…We overcome these difficulties by utilising the O(t m )-weighted L 2 estimates of the mth-order time derivative and 2mth-order spatial derivatives (as shown in Lemma 3.2) and a duality argument with variable temporal stepsizes to resolve the initial singularity in the consistency errors (as shown in Section 3.3). It is known that variable stepsizes can help resolve the singularity in proving convergence of exponential integrators for semilinear parabolic equations with nonsmooth initial data; see [33]. However, the error analysis for the NS equations turns out to be completely different from the error analysis for the semilinear parabolic equation due to the lack of Lipschitz continuity of the nonlinearity and the critical nature of the L2 space.…”

Analysis of fully discrete finite element methods for 2D Navier--Stokes equations with critical initial data

“…The development of efficient numerical methods that may have some uniform temporal convergence up to t = 0 is still challenging. In view of the low-regularity integrators recently developed for dispersive equations [25,34,36] and semilinear parabolic equations [32] this is possible and worth to be considered (at least for semi-discretization in time).…”

“…We overcome these difficulties by utilising the O(t m )-weighted L 2 estimates of the mth-order time derivative and 2mth-order spatial derivatives (as shown in Lemma 3.2) and a duality argument with variable temporal stepsizes to resolve the initial singularity in the consistency errors (as shown in Section 3.3). It is known that variable stepsizes can help resolve the singularity in proving convergence of exponential integrators for semilinear parabolic equations with nonsmooth initial data; see [32]. However, the error analysis for the NS equations turns out to be completely different from the error analysis for the semilinear parabolic equation due to the lack of Lipschitz continuity of the nonlinearity and the critical nature of the L2 space.…”

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

“…Therefore, the key consideration in solving the reaction-subdiffusion equations is how to resolve the initial singularity behavior. One simple and flexible way to deal with the initial singularity among many existing methods is to use nonuniform time steps (see [12,13,15,34]). For these reasons, numerical schemes on nonuniform meshes have attracted increasing attention.…”