We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on f log f as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on 1 2f as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.The porous medium equation (PME) can be found in many physical and biological phenomena, such as the flow of an isentropic gas through a porous medium [18], the viscous gravity currents [12], nonlinear heat transfer and image processing; e.g., see [33]. The aim of this paper is to provide numerical methods for the PMEwhere f := f (x, t) is a non-negative scalar function of space x ∈ R d and the time t ∈ R, the space dimension is given by d ≥ 1, and m is a constant larger than 1.The PME is a nonlinear degenerate parabolic equation since the diffusivity D(f ) = mf m−1 = 0 at points where f = 0. In turn, the PME has a special feature: the finite speed of propagation, called finite propagation [33]. If the initial data has a compact support, the solution of Cauchy problem of the PME will have a compact support at any given time t > 0. In comparison with the heat equation, which can smooth out the initial data, the solution of the PME becomes non-smooth even if the initial data is smooth with compact support. If an initial data is zero in some open domain in Ω, it causes the appearance of the free boundary (in some cases, called interface) that separates the regions where the solution is positive from the regions where the value is zero in the domain. Moreover, for certain initial data, the solution of the PME can exhibit a waiting time phenomenon where the free boundary remains stationary until a finite positive time (called waiting time).After that time instant, the interface begins to move with a finite speed. Many theoretical analyses have been available in the existing literature, including the earlier works by Oleǐnik et al. [25], Ka...

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on f log f as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞ norm, and a refined estimate is applied to derive the optimal error order.One of the typical nonlinear degenerate parabolic equations is the porous medium equation (PME):and the time t ∈ R + , and m is a constant larger than 1. It has been applied in many physical and biological models, such as an isentropic gas flow through a porous medium, the viscous gravity currents, nonlinear heat transfer and image processing [18], etc.It is well known that the PME is degenerate at points where f = 0. In turn, the PME has many special features: the finite speed of propagation, the free boundary, a possible waiting time phenomenon [5,18]. Various numerical methods have been studied for the PME, such as finite difference approach [8], tracking algorithm method [3], a local discontinuous Galerkin finite element method [24], Variational Particle Scheme (VPS) [23] and an adaptive moving mesh finite element method [13]. Many theoretical analyses have been derived in the existing literature [1,12,14,16,17,18], etc.Relevant detailed descriptions can be found in a recent paper [5], in which the numerical methods for the PME were constructed by an Energetic Variational Approach (EnVarA) to naturally keep the physical laws, such as the conservation of mass, energy dissipation and force balance. Meanwhile, based on different dissipative energy laws, two different numerical schemes have been studied. In more details, based on the total energy form f log f and 1 2f , a fully discrete nonlinear scheme and a linear numerical scheme could be appropriately designed for the trajectory equation, respectively. It has also been proved that the former one is uniquely solvable on an admissible convex set, and both schemes preserve the corresponding discrete dissipation law. Numerical experiments have demonstrated that both schemes have yielded a good approximation for the solution without oscillation and the free boundary. The notable advantage is that the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existing methods. In ad...

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