A transformed stochastic Euler scheme for multidimensional transmission PDE

TL;DR

This paper introduces a low-complexity Monte Carlo Euler scheme for solving multidimensional parabolic PDEs with discontinuous coefficients across interfaces, providing convergence guarantees and accuracy comparisons.

Contribution

It develops a novel stochastic Euler scheme for multidimensional transmission PDEs with discontinuous coefficients, including new derivative estimates and convergence analysis.

Findings

The scheme achieves the same convergence rate as 1D stochastic methods.

It provides new estimates for solution derivatives.

Numerical results confirm the method's accuracy.

Abstract

In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is assumed to present a compatibility transmission condition of its conormal derivatives at this interface (multi-dimensional diffraction problem). We prove an existence and uniqueness result for the solution and construct a low complexity numerical Monte Carlo stochastic Euler scheme to approximate the solution of the parabolic partial differential equation in divergence form. In particular, we give new estimates for the partial derivatives of the solution. Using these estimates, we prove a convergence rate for our stochastic numerical method when the initial condition belongs to an iterated domain of the divergence form operator. Our method presents the same convergence rate as the stochastic numerical schemes elaborated for the same problem in the one-dimensional context. Finally, we compare our results to classical deterministic numerical approximations and illustrate the accuracy of our method.