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A new class of time discretization schemes for the solution of nonlinear PDEs. (English) Zbl 0924.65089
Stability of time-discretization schemes with exact treatment of the linear part (ELP schemes) for solving nonlinear partial differential equations (PDEs) is considered. ELP schemes are based on exact evaluation of the linear term. Computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient. Using computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices which is very effective from the numerical point of view. This fact is used for computing functions of elliptic operators with variable coefficients.
In this paper this approach is developed on issues of stability and it is shown that both explicit and implicit ELP schemes have distinctly different stability properties as compared with known implicit-expilcit schemes. Examples of nonlinear diffusion equation with forcing term and Burger equation are computed using explicit ELP schemes at the end of this paper.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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