High accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients
In the region \((0,1)^2 \times (0,\infty)\) the authors consider the following system of nonlinear parabolic equations: \[ A(x,y,t) u_{xx} +2B(x,y,t) u_{xy} +C(x,y,t) u_{yy} =f(x,y,t,u^T, u^T_x, u^T_y, u^T_t), \] subject to initial and Dirichlet boundary conditions, where \[ u = [u^{(1)}, \dots, u^{(M)}]^T,\quad f=[f^{(1)}, \dots, f^{(M)}]^T,\quad u_{xx} =[u^{(1)}_{xx}, \dots, u_{xx}^{ (M)}]^T, \] etc. and \(A,B,C\) are \(M \times M\) diagonal matrices of the form \(\text{diag} [S^1(x,y, t), \dots, S^{(M)} (x,y,t)]\) for \(S=A,B\) and \(C\), respectively. They examine two cases: \(A^{(i)}= C^{(i)}\), \(B^{(i)} \neq 0\) or \(B^{(i)} =0\). Assuming \(A^{(i)} >0\), \(C^{(i)} >0\), \((B^{(i)})^2 <A^{(i)} C^{(i)}\) and suitable regularity conditions on \(u^{(i)}\), \(A^{(i)}\), \(B^{(i)}\), \(C^{(i)}\) they propose, using 9-spatial grid points, a two-level implicit finite difference scheme for the above problem. In an appendix a derivation of this finite difference method is given in details. The accuracy of the method is \(O(k^2+ kh^2+ h^4)\), where \(h\) and \(k\) are the step lengths in space and time directions, respectively. The difference method is applied (after some modification) to a linear scalar parabolic equation. Next, a series of problems (heat equation, two-space unsteady Navier-Stokes equations), both linear and nonlinear, are formulated in polar, spherical and cylindrical coordinates and treated by the authors' method, which turns out to be an unconditionally stable ADI method of \(O(k^2+ h^4)\) accuracy, also in the vicinity of \(r=0\) and even for \(r=0\). Computational implementations for some examples are tabulated.
scientific article; zbMATH DE number 908425