Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

TL;DR

This paper introduces a versatile class of semi-Lagrangian schemes for linear and non-linear diffusion equations, capable of handling complex coefficient matrices and providing efficient, stable, and convergent numerical solutions.

Contribution

It presents a unifying framework for existing schemes and introduces new higher order methods that are easy to implement and more efficient.

Findings

Schemes are consistent and stable.

Monotone first order schemes converge generally.

Numerical tests confirm efficiency and robustness.

Abstract

For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.