Related papers: A positivity-preserving and energy stable scheme f…
We propose a new numerical scheme of evolution for the Einstein equations using the discrete variational derivative method (DVDM). We derive the discrete evolution equation of the constraint using this scheme and show the constraint…
We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form elliptic equations. Based on the Feynman--Kac formula, the solution is represented as a conditional expectation associated with a diffusion…
Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic…
We consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution…
In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical ($\alpha=1$) differential equation in the fractional case (here using fractional…
We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be…
We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under {\sl anisotropic surface diffusion} with a general anisotropic surface energy $\gamma(\boldsymbol{n})$…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…
This paper introduces improved numerical techniques for addressing numerical boundary and interface coupling conditions in the context of diffusion equations in cellular biophysics or heat conduction problems in fluid-structure…
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the…
We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. The scheme preserves two important structural properties of the…
Upon compressing perceptually relevant signals, conventional quantization generally results in unnatural outcomes at low rates. We propose distribution preserving quantization (DPQ) to solve this problem. DPQ is a new quantization concept…
Hybrid finite element methods, particularly hybridized discontinuous Galerkin (HDG) methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical principles: mass conservation and…
We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature…
In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an…