Related papers: Implicit like time discretization for the one-phas…
In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid…
Nonlinear time-dependent differential equations for the Hele-Shaw, Saffman-Taylor problem are derived. The equations are obtained using a separable ansatz expansion for the stream function of the displaced fluid obeying a Darcian flow.…
In this article, we apply the viscosity solutions theory for integro-differential equations to the \emph{one-phase} Muskat equation (also known as the Hele-Shaw problem with gravity). We prove global well-posedness for the corresponding…
This paper addresses a rescaling behavior of some classes of global solutions to the zero surface tension Hele-Shaw problem with injection at the origin, $\{\Omega(t)\}_{t\geq 0}$. Here $\Omega(0)$ is a small perturbation of $f(B_{1}(0),0)$…
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only…
We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step…
In this paper we study the numerical error arising in the space-time approximation of unsteady generalized Newtonian fluids which possess a stress-tensor with $(p,\delta)$-structure. A semi-implicit time-discretization scheme coupled with…
Surface tension has a strong influence on the shape of fluid interfaces. We propose a method to calculate the corresponding forces efficiently. In contrast to several previous approaches, we discriminate to this end between surface and…
The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic…
We study a family of solutions to the Saffman-Taylor problem with zero surface tension at a critical regime. In this regime, the interface develops a thin singular finger. The flow of an isolated finger is given by the Whitham equations for…
The present paper deals with the interior solid-fluid interaction problem in harmonic regime with randomly perturbed boundaries. Analysis of the shape derivative and shape Hessian of vector- and tensor-valued functions is provided. Moments…
A one-sided phase-field model is proposed to study the dynamics of unstable interfaces of Hele-Shaw flows in the high viscosity contrast regime. The corresponding macroscopic equations are obtained by means of an asymptotic expansion from…
In this paper, we give a precise description of the rescaling behaviors of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of…
We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele-Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof…
For the one-dimensional Helmholtz equation we write the corresponding time-dependent Helmholtz Hamiltonian in order to study it as an Ermakov problem and derive geometrical angles and phases in this context
We study the two dimensional water waves problem with surface tension in the case when there is a non-zero contact angle between the free surface and the bottom. In the presence of surface tension, dissipations take place at the contact…
We investigate the dynamics of relaxation, by surface tension, of a family of curved interfaces between an inviscid and viscous fluids in a Hele-Shaw cell. At t=0 the interface is assumed to be of the form |y|=A x^m, where A>0, m \geq 0,…
The time-dependent surface flux method developed for the description of electronic spectra [L. Tao and A. Scrinzi, New J. Phys. 14, 013021 (2012); A. Scrinzi, New J. Phys. 14, 085008 (2012)] is extended to treat dissociation and…
In this paper, we present the numerical solution of two-phase flow problems of engineering significance with a space-time finite element method that allows for local temporal refinement. Our basis is the method presented in [3], which…
We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of a SDE with a time changed Brownian motion, dated back to Doeblin…