Difference schemes with point symmetries and their numerical tests

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure

Numerical investigations of traveling singular sources problems via moving mesh method

This paper studies the numerical solution of traveling singular sources problems. In such problems, a big challenge is the sources move with different speeds, which are described by some ordinary differential equations. A predictor-corrector algorithm is presented to simulate the position of singular sources. Then a moving mesh method in conjunction with domain decomposition is derived for the underlying PDE. According to the positions of the sources, the whole domain is splitted into several subdomains, where moving mesh equations are solved respectively. On the resulting mesh, the computation of jump $[\dot{u}]$ is avoided and the discretization of the underlying PDE is reduced into only two cases. In addition, the new method has a desired second-order of the spatial convergence. Numerical examples are presented to illustrate the convergence rates and the efficiency of the method. Blow-up phenomenon is also investigated for various motions of the sources

A blob method for diffusion

As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles, and we apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker-Planck equation, the Keller-Segel equation, and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method, again under sufficient regularity assumptions. We conclude by considering a range of numerical examples to demonstrate our method's rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation