On entropy satisfying and maximum-principle-satisfying high-order methods for Fokker-Planck equations
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Computation of Fokker-Planck equations with satisfying long time behavior is important in many applications. In this thesis, we design, analyze and implement entropy satisfying and maximum-principle-satisfying high-order numerical methods to solve the Fokker-Planck equation of the finitely extensible nonlinear elastic (FENE) dumbbell model for polymers, subject to homogeneous fluids, and the reaction-diffusion-advection equation arising in the evolution of biased dispersal of population dynamics. The design of each method is guided to satisfy three main properties, consisting of the nonnegativity principle, the mass conservation and the preservation of nonzero steady states. The relative entropy and the maximum principle are two powerful tools used to evaluate our methods, for instance, the steady state preservation can be ensured if the method is either entropy satisfying or maximum principle satisfying in the sense that the ratio of the solution to the equilibrium will stay in the same range as indicated by the initial data.
These schemes are constructed in several steps, including reformulation of the Fokker-Planck equation into its nonlogarithmic Landau form, spacial discretization by discontinuous Galerken (DG) methods and some Runge-Kutta time discretization. The special form of numerical fluxes motivated by those introduced in [H. Liu and J. Yan, Commun. Comput. Phys. 8(3), 2010, 541-564] is essential to incorporate desired properties into each scheme through choices of flux parameters.
In this thesis, we have obtained the following results.
1. For the Fokker-Planck equation of the FENE model, we propose an entropy satisfying conservative method which preserves all the three desired properties at both semidiscrete and discrete levels. This method is shown to be entropy satisfying in the sense that these schemes satisfy discrete entropy inequalities for both the physical entropy and the quadratic entropy. These ensure that the computed solution is a probability density, and the schemes are entropy stable and preserve the equilibrium solutions.
2. We further develop an entropy satisfying DG method of arbitrary high order. Both semidiscrete and fully discrete methods are shown to satisfy two desired properties: mass conservation and entropy satisfying for the quadratic entropy, therefore preserving the equilibrium solutions. A positive numerical approximation is obtained with the same accuracy as the numerical solution through a reconstruction at the final time.
For both the finite volume scheme and the DG scheme we also prove the convergence of numerical solutions to the equilibrium solution as time tends to infinity. One- and two-dimensional numerical results are provided to demonstrate the good qualities of these schemes and effects of some canonical homogeneous flows.
3. We develop up to third-order accurate DG methods satisfying a strict maximum principle for a class of linear Fokker-Planck equations. A procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution. This is achievable by properly choosing flux parameters and a positive decomposition of weighted cell averages. Based on this result, a scaling limiter for the DG method with Euler forward time discretization is proposed to solve both one- and multidimensional Fokker-Planck equations. As a consequence, the present scheme preserves steady states and provides a satisfying long time behavior. Numerical tests for the DG method are reported, with applications to polymer models with both Hookean and FENE potentials.
4. For Fokker-Planck equations with reaction such as the biased dispersal model in population dynamics, we develop entropy/energy stable finite difference schemes. For the numerical method to capture the long-time pattern of persistence or extinction, we use the relative entropy when the resource potential is logarithmic and explore the usual energy for other resource potentials. The present schemes are shown to satisfy three important properties of the continuous model for the population density: (i) positivity preserving, (ii) equilibrium preserving and (iii) entropy or energy satisfying. These ensure that our schemes provide a satisfying long-time behavior, thus revealing the desired dispersal pattern. Moreover, we present several numerical results which confirm the second-order accuracy for various resource potentials and underline the efficiency to preserve the large time asymptotic.