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In this note we develop a numerical method for partial differential equations with changing type. Our method is based on a unified solution theory found by Rainer Picard for several linear equations from mathematical physics. Parallel to…
A numerical method based on the hybridizable discontinuous Galerkin method in space and backward Euler in time is formulated and analyzed for solving the miscible displacement problem. Under low regularity assumptions, convergence is…
Heterogeneous materials, such as rocks and concrete, have a complex dynamics including hysteresis, nonlinear elasticity and viscoelasticity. It is very sensitive to microstructural changes and damage. The goal of this paper is to propose a…
In this paper, we investigate the numerical solutions of the cubic nonlinear Schrodinger equation via the exponential B-spline collocation method. Crank-Nicolson formulas are used for time discretization of the target equation. A…
A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are…
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system…
We develop an approach to solving numerically the time-dependent Schrodinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an…
This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well-posedness are derived in terms of the material parameters. The…
Dual-continuum (DC) models can be tractable alternatives to explicit approaches for the numerical modelling of multiscale materials with multiphysics behaviours. This work concerns the conceptual and numerical modelling of poroelastically…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
The development of efficient and robust dynamic models is fundamental in the field of systems and control engineering. In this paper, a new formulation for the dynamic model of nonlinear mechanical systems, that can be applied to different…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. The models take the form of compressible Navier-Stokes equations whose pressure is determined by the…
We present a numerical framework for modeling extended hyperelastic bodies based on a Lagrangian formulation of general relativistic elasticity theory. We use finite element methods to discretize the body, then use the semi--discrete action…
We demonstrate that the machine learning of density functionals allows one to determine simultaneously the equilibrium chemical potential across simulation datasets of inhomogeneous classical fluids. Minimization of a loss function based on…
In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second…
One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
The linear response of an isolated, homogeneous granular fluid to small spatial perturbations is studied by methods of non-equilibrium statistical mechanics. The long wavelength linear hydrodynamic equations are obtained, with formally…