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A meshless pseudospectral approach appliedto problems with weak discontinuities

Krowiak Artur

Computer Assisted Methods in Engineering and Science

2022

Vol. 29, no. 3

179--195

In this paper, a meshless pseudospectral method is applied to solve problems possessingweak discontinuities on interfaces. To discretize a differential problem, a global inter-polation by radial basis functions is used with the collocation procedure. This leads toobtaining the differentiation matrix for the global approximation of the differential opera-tor from the analyzed equation. Using this matrix, the discretization of the problem isstraightforward. To deal with the differential equations with discontinuous coefficients onthe interface, the meshless pseudospectral formulation is used with the so-called subdo-main approach, where proper continuity conditions are used to obtain accurate results.In the present paper, the differentiation matrix for this method is derived and the choiceof a proper value of the shape parameter for radial functions in the context of the subdo-main approach is studied. The numerical tests show the effectiveness of the method whenusing regular or unstructured node distribution. They confirm that the approach preserveswell-known advantages of the radial basis function collocation method, i.e., rapid conver-gence, simplicity of the implementation and extends its usage for problems with weakdiscontinuity.

Initial value problem for the time-dependent linear Schrodinger equation with a point singular potential by the unified transform method

Rybalko Y.

Opuscula Mathematica

2018

Vol. 38, no. 6

883--898

We study an initial value problem for the one-dimensional non-stationary linear Schrodinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem.