Semi-analytic time-marching algorithms for semilinear parabolic equations
New time marching algorithms for the numerical solution of semilinear parabolic equations are described. They are based on the approximation method proposed by the first author (based upon the use of approximate partition of unity). The algorithms developed in this paper give the time step approximation in the form of an analytic function, which enables one to differentiate the approximate solution with respect to \(x\) explicitly. The algorithms are flexible in the choice of the basis functions and they can be generalized to the multidimensional case easier than spline approximations and other finite element procedures. An important advantage is the possibility to obtain explicit formulae for values of various integral and pseudodifferential operators of mathematical physics applied to the basis functions. The algorithms are stable under mild restrictions to the time step, which come from the nonlinear part of the equation.
scientific article; zbMATH DE number 716132