O(h4) locally overconvergent semidiscrete scheme for the equation ut = uxx + f(t, x, u)

摘要

In the equation ut = uxx + f(t, x, u) the second derivative uxx is approximated by the finite-difference operator Lh which has O(h2) local truncation error at the points h and 1 − h next to the ends of the interval [0, 1] and O(h4) at the interior mesh points 2h, 3h,…, 1 − 2h. It is proved that the obtained semidiscrete scheme is O(h4) globally convergent. The semidiscrete scheme is solved by the pure implicit finite-difference scheme combined with the method of iteration and the Gauss elimination method for tridiagonal matrices. The nonstationary Liouville's equation ut = uxx + e−u has been solved by the proposed algorithm.