\(O(\tau^ 2 + h^ 4)\) finite difference scheme for decoupled system of two quasilinear parabolic equations (Q1919398)

For \(0<x<1\), \(t>0\) the author considers the following system of equations: \[ p_i(t,x) {\partial u_i \over\partial t} (t,x)= a_i(t,x) {\partial^2 u_i \over\partial x^2} (t,x)+ f_i(t,x, u_1(t,x), u_2(t,x)),\quad i=1,2, \] together with initial and homogeneous Dirichlet boundary conditions under appropriate regularity assumptions on \(p_i\), \(a_i\), \(f_i\), \(i=1,2\). Applying a five-points finite difference operator to \({\partial^2u_i \over \partial x^2} (i=1,2)\) in mesh points \(x_j =jh\) \((h>0)\), \(j=1, \dots, N-1\), he obtains the system of ordinary differential equations. He proves that this method of lines is globally convergent having the error \(O(h^4)\). Next, the trapezoidal rule is used for obtaining a fully discrete scheme, which is solved by an implicit iterative process. Numerical results are given for \(f_1= \exp (-u_1 +u_2)\), \(f_2= \exp (u_1-u_2)\).