Related papers: Stability and error analysis for a diffuse interfa…
The evolution of interfaces is intrinsic to many physical processes ranging from cavitation in fluids to recrystallization in solids. Computational modeling of interface motion entails a number of challenges, many of which are related to…
Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media,…
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,…
This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear…
We design, analyse and implement an arbitrary order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretisation is based on several adaptations of the…
We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value.…
This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf-sup stable discretization schemes for a…
We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high…
In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in…
A general high-order fully explicit scheme based on projective integration methods is here presented to solve systems of degenerate parabolic equations in general dimensions. The method is based on a BGK approximation of the…
We analyze from a far field the evolution of an interface that separates ideal incompressible fluids of different densities and has an interfacial mass flux. We develop and apply the general matrix method to rigorously solve the boundary…
In this paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Such systems of coupled partial differential equations arise in, for example, the modeling…
The interface stability against small perturbations of the planar solid-liquid interface is considered analytically in linear approximation. Following the analytical procedure of Trivedi and Kurz (Trivedi R, Kurz W. Acta Metall…
This article is concerned with the development of a theoretical framework of global measure-valued solutions for a class of hyperbolic-parabolic cross-diffusion systems, and its application to the convergence analysis of a fully discrete…
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
In this work, we consider a certain multilayered (thick layer) wave--(thin layer) wave--heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian…
We present a method to simulate fluid flow on evolving surfaces, e.g., an oil film on a water surface. Given an animated surface (e.g., extracted from a particle-based fluid simulation) in three-dimensional space, we add a second simulation…
The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach has been studied in a recent work [6], in which the trajectory equation is obtained, and a few first order accurate…
In this work, we study the well-posedness of a system of partial differential equations that model the dynamics of a two-dimensional Stokes bubble immersed in two-dimensional ambient Stokes fluid of the same viscosity that extends to…
We study a model describing the slow flow of a fluid through a deformable, porous, elastic solid undergoing small deformations. The stress-strain relationship of the solid incorporates nonlinear effects, formulated as a perturbation of the…