Related papers: Exact domain truncation for the Morse-Ingard equat…
In 1957, Mullins proposed surface diffusion motion as a model for thermal grooving. By adopting a small slope approximation, he reduced the model to the Mullins' linear surface diffusion equation, \begin{equation} \nonumber…
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their…
Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential…
We consider wave propagation in a coupled fluid-solid region, separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic…
We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We…
We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of…
Finite-dimensional truncations are routinely used to approximate partial differential equations (PDEs), either to obtain numerical solutions or to derive reduced-order models. The resulting discretized equations are known to violate certain…
The numerical approximation of high-frequency wave propagation in inhomogeneous media is a challenging problem. In particular, computing high-frequency solutions by direct simulations requires several points per wavelength for stability and…
We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to…
Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure…
This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation…
We present strongly stable semi-discrete finite difference approximations to the quarter space problem (x>0, t>0) for the first order in time, second order in space wave equation with a shift term. We consider space-like (pure outflow) and…
In this paper an exact transparent boundary condition (TBC) for the multidimensional Schr\"odinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions…
The Jordan--Moore--Gibson--Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third-order (in time) semilinear Partial Differential Equation (PDE) model with the…
The linearized, compressible Navier-Stokes equations can be used to model acoustic wave propagation in the presence of viscous and thermal boundary layers. However, acoustic boundary layers are notorious for invoking prohibitively high…
We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved…
Based on a unique waveform with strong exponential localization property, an exact mathematical method for solving problems in signal analysis in time-frequency domain is presented. An analogue of the Gabor frame exposes the non-commutative…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…
The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…
We study various formulations of the boundary conditions for the Euler equations of gas dynamics from a mathematical and numerical point of view. In the case of one space dimension, we recall the classical results, based on an analysis of…