Spectrum-free and meshless solvers of parabolic PDEs
The paper presents a collocation method for solving parabolic partial differential equations, e.g., the inhomogeneous heat equation, in two and three space dimensions. In space, the solution is discretized into a series of radial basis functions, e.g., Gaussian kernels or multi-quadratic kernels. For time integration,the semi-group of the heat equation is used and the corresponding exponential integrator is discretized with a Padé-Chebychev method. Moreover, for comparison, a linear combination of the implicit and explicit Euler method is used, i.e., a so-called \(\theta\) method, which corresponds to the Crank-Nicolson method for \(\theta=1/2\). Convergence of the time integration using the Padé-Chebychev method is provided and the existence of the spatial derivatives of the radial basis functions is analyzed. Then numerical properties and test computations are carried out and the computational cost is assessed. Convection-diffusion problems are likewise studied. Numerical results in 2D and 3D are reported for regular and irregular sampling in space and the errors with respect to a reference solution are determined. It is shown that the Padé-Chebychev method yields higher stability and better accuracy compared to previous results in the literature.
scientific article; zbMATH DE number 7388734
scientific article