Justification of difference schemes for derivative nonlinear evolution equations

Abstract

We consider the difference schemes applied to a derivative nonlinear system of evolution equations. For the boundary-value problem with initial conditions

$$\begin{gathered}\frac{{\partial u}}{{\partial t}} = A\frac{{\partial ^2 u}}{{\partial x^2 }} + B\frac{{\partial u}}{{\partial x}} + f(x,u) + g(x,u)\frac{{\partial u}}{{\partial x}},(t,x) \in (0,T] \times (0,1), \hfill \\u(t,0) = u(t,1) = 0,t \in [0,T], \hfill \\u(0,x) = u^{(0)} (x),x \in (0,1) \hfill \\\end{gathered}$$

we use the Crank-Nicolson discretizations. A is complex and B — real diagonal matrixes; u,f and g are complex vector-functions. The analysis shows that proposed schemes are uniquely solvable, convergent and stable in a grid norm W ²₂ if all (diagonal) elements in Re(A) are positive. This is true without any restrictions on the ratio of time and space grid steps.