Role of parabolic viscosity in numerical analysis of 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

∂u/∂t = A ∂²u/∂x² + B ∂u/∂x + f(x,u) + g(x,u) ∂u/∂x,    (t,x) ∈  (0, T] x (0, 1),
u(t,0) = u(t,1) = 0,    t ∈ [0, T],
u(0,x) = u⁽⁰⁾(x),    x ∈ (0, 1)

we use the Crank-Nicolson discretization. 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 W₂² if all (diagonal) elements in Re(A) are positive. This is true without any restriction on the ratio of time and space grid steps.