Semi-Lagrangian schemes for linear and fully non-linear diffusion equations (Q2840616)

There is an attempt to analyse a class of approximation schemes for fully nonlinear Hamilton-Jacobi-Bellman equations. The authors introduce two schemes for the solution of the problems: (i) a linear semi Lagrangian (LISL)scheme defined on unstructured grids and (ii) a cubic interpolation semi-Lagrangian scheme (MPCSL) based on Hermite cubic interpolation. The time discretization is done through a mid-point rule that includes explicit, implicit and a second order Crank-Nicolson approximation. Some examples are considered to demonstrate the usefulness of the schemes. The MPCSL scheme has demonstrated better in reducing the error propagations as compared to the LISL scheme for a linear problem with smooth solution which is obvious because the third-order monotinicity is preserved in the first scheme. The error propagation is comparable in both schemes for the nonsmooth linear problems. Also the MPCSL scheme performs better than the LISL scheme for optimal control problems. Although the CPU times are comparable for both schemes, the order of convergence is better in the MPCSL scheme. Several lemmas and theorems are proved in developing these two schemes.