Role of parabolic viscosity in numerical analysis of derivative nonlinear evolution equations
Articles
Tadas Meškauskas
Vilnius University, Lithuania
Feliksas Ivanauskas
Vilnius University, Lithuania
Published 1998-12-21
https://doi.org/10.15388/NA.1998.2.0.15288
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How to Cite

Meškauskas T. and Ivanauskas F. (1998) “Role of parabolic viscosity in numerical analysis of derivative nonlinear evolution equations ”, Nonlinear Analysis: Modelling and Control, 2(1), pp. 75-80. doi: 10.15388/NA.1998.2.0.15288.

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.

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