MATH-PHMay 30, 2018
Compact difference scheme for parabolic and Schrödinger-type equations with variable coefficientsVladimir Gordin, Evgenii Tsymbalov
We develop a new compact scheme for second-order PDE (parabolic and Schrödinger type) with a variable time-independent coefficient. It has a higher order and smaller error than classic implicit scheme. The Dirichlet and Neumann boundary problems are considered. The relative finite-difference operator is almost self-adjoint.
MATH-PHDec 27, 2018
Compact difference schemes for weakly-nonlinear parabolic and Schrodinger-type equations and systemsVladimir Gordin, Evgenii Tsymbalov
The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schrödinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we need to solve a non-linear system. We use for this step a simple explicit difference scheme and then Newton -- Raphson iterations, which are implemented by the double-sweep method. Numerical experiments confirm the 4-th order of an algorithm. The Richardson extrapolation improves it up to the 6-th order.