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The purpose of this paper is the numerical analysis of a first order fractional-step time-scheme, using decomposition of theviscosity, and "inf-sup" stable finite element space-approximations for the Primitive Equations of the Ocean. The…
Estimation of the degree of stability and the bounds of solutions to non-autonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are…
Solutions of the Dirichlet and Robin boundary value problems for the multi-term variable-distributed order diffusion equation are studied. A priori estimates for the corresponding differential and difference problems are obtained by using…
We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The…
We construct numerical schemes to solve kinetic equations with anomalous diffusion scaling. When the equilibrium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling should be made and the…
This paper discusses the initial-boundary value problem (with a nonhomogeneous boundary condition) for a multi-dimensional scalar first-order conservation law with a multiplicative noise. One introduces a notion of kinetic formulations in…
We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of…
We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces,…
We introduce an adaptive viscosity regularization approach for the numerical solution of systems of nonlinear conservation laws with shock waves. The approach seeks to solve a sequence of regularized problems consisting of the system of…
We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial…
We study the inflow-outflow boundary value problem on an interval, the analog of the 1D shock tube problem for gas dynamics, for general systems of hyperbolic-parabolic conservation laws. In a first set of investigations, we study…
In this paper, approximation schemes are proposed for handling load uncertainty in compliance-based topology optimization problems, where the uncertainty is described in the form of a set of finitely many loading scenarios. Efficient…
We present a numerical scheme for the solution of the initial-value problem for the ``bad'' Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic…
We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy…
We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and…
Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in…
This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The…
We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a…
We illustrate that numerical solutions of high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for some nonconvex conservation laws perform poorly or converge to the entropy solution in a slow speed. The…
We introduce a finite volume scheme to solve a special case of isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions…