Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit

Numerical Studies of Discrete Approximations to the Allen--Cahn Equation in the Sharp Interface Limit

The numerical approximations to the Allen–Cahn type diffuse interface models are studied, with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes. Different spatial discretizations of an energy functional in the diffuse interface framework are compared first. Discretizations of the time-dependent equation using various...

Sharp interface limit of the Fisher-KPP equation

We investigate the singular limit, as ε → 0, of the Fisher equation ∂tu = ε∆u+ ε u(1− u) in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as ‖x‖ → ∞. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional t...

Sharp Interface Limit for Invariant Measures of a Stochastic Allen-cahn Equation

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density which can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs type measure on a growing interval with ...

On Numerical Approximation of Stochastic Burgers’ Equation

Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreo...