An Euler–Maruyama method for diffusion equations with discontinuous coefficients and a family of interface conditions

Numerical methods are developed for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients along with a one parameter family of interface conditions at the discontinuity. We construct an Euler–Maruyama numerical method for the stochastic differential equation (SDE) corresponding to the alternative divergence formulation of these equations. Our main result is the construction of an Euler scheme that can accommodate specification of any one of a family of interface conditions considered. We then prove convergence estimates for the Euler scheme. To illustrate our method and its theoretical analysis we implement it for the stochastic formulation of the parabolic system.