The article discusses a method for the numerical approximation of nonlinear SPDEs with additive noise which allow for a mild solution. In the presented setting, the nonlinearity satisfies standard global Lipschitz assumptions. The numerical method is an extension of the scheme developed by A. Jentzen and P. E. Kloeden [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, No. 2102, 649–667 (2009; Zbl 1186.65011)] to the present class of SPDEs. It is an exponential Euler scheme using suitable functionals of the noise process which yield an improvement in the order of convergence. It is noted that the applicability of the method is restricted to SPDEs for which the eigenfunctions of the linear operator and the covariance operator of the driving noise process coincide and are explicitly known.
The authors prove the strong convergence of this problem, i.e., the mean-square convergence at the endpoint of a time interval in the appropriate Hilbert space of the solution, and obtain an expression for the order of convergence. The advantage of the presented method is that it is asymptotically more efficient than, e.g., the linear implicit Euler method, i.e., a given precision is obtained with a smaller number of computational operations and independent standard normal random variables. In particular, the new method possesses an effort of compared to of the linear-implicit Euler method.
Numerical examples are presented to illustrate the theoretical results: the improvement in convergence order of the new method as well as its higher efficiency. For comparison, a linear implicit Euler-spectral Galerkin method is used. The numerical examples are simple reaction diffusion equations on the one-dimensional unit interval with additive noise and twice continuously differentiable nonlinearity with bounded derivatives. Finally, a numerical example with nonglobally Lipschitz continuous nonlinearity is implemented and the pathwise error of the method is calculated. Again, a higher convergence order is experimentally found compared to the linear-implicit Euler method.