The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations. Two applications are investigated, linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication-type equations for which it is unclear if the convergence analysis applies. The numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature. The convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions. Cutoff can also be used with other discretization methods such as collocation, finite volume, finite element, and spectral methods and for the computation of positive solutions.Comment: 19 pages, 41 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

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

Space-time domain decomposition for advection-diffusion problems in mixed formulations

This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time steps can be used in different parts of the domain. Global-in-time, non-overlapping domain-decomposition methods are coupled with operator splitting making possible the different treatment of the advection and diffusion terms. Two domain-decomposition methods are considered: one uses the time-dependent Steklov--Poincar{\'e} operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains. A generalized Neumann-Neumann preconditioner is proposed for the first method. To illustrate the two methods numerical results for two-dimensional problems with strong heterogeneities are presented. These include both academic problems and more realistic prototypes for simulations for the underground storage of nuclear waste

On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws

We prove convergence of a class of space-time discontinuous Galerkin schemes for scalar hyperbolic conservation laws. Convergence to the unique entropy solution is shown for all orders of polynomial approximation, provided strictly monotone flux functions and a suitable shock-capturing operator are used. The main improvement, compared to previously published results of similar scope, is that no streamline-diffusion stabilization is used. This is the way discontinuous Galerkin schemes were originally proposed, and are most often used in practice