Steady state spurious errors in shock-capturing numerical schemes

The behavior of the steady state spurious error modes of the MacCormack scheme and the upwind scheme of Warming and Beam was obtained from a linearized difference equation for the steady state error. It was shown that the spurious errors can exist either as an eigensolution of the homogeneous part of this difference equation or because of excitation from large discretization errors near oblique shocks. It was found that the upwind scheme does not permit spurious oscillations on the upstream side of shocks. Examples are given for the inviscid Burgers' equation and for one and two dimensional gasdynamic flows

A convergent nonconforming finite element method for compressible Stokes flow

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div-curl form using the nonconforming Crouzeix-Raviart space. Our main result is that the finite element method converges to a weak solution. The main challenge is to demonstrate the strong convergence of the density approximations, which is mandatory in view of the nonlinear pressure function. The analysis makes use of a higher integrability estimate on the density approximations, an equation for the "effective viscous flux", and renormalized versions of the discontinuous Galerkin method.Comment: 23 page

Material Issues of the Metal Printing Process, MPP

The metal printing process, MPP; is a novel Rapid Manufacturing process under development at SINTEF and NTNU in Trondheim, Norway. The process, which aims at the manufacturing of end-use products for demanding applications in metallic and CerMet materials, consists of two separate parts; The layer fabrication, based on electrostatic attraction of powder materials, and the consolidation, consisting of the compression and sintering of each layer in a heated die. This approach leads to a number of issues regarding the interaction between the process solutions and the materials. This paper addresses some of the most critical material issues at the current development stage of MPP, and the present solutions to these.Mechanical Engineerin

Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial-boundary value problem. The proof utilizes the kinetic formulation and the compensated compactness method. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial-boundary value problem for the Degasperis-Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.Comment: 24 page

Well-posedness results for triply nonlinear degenerate parabolic equations

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0$$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and $\psi$ are supposed to be continuous non-decreasing, and the nonlinearity $\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\tilde{\mathfrak a}(u,\nabla\phi(u))$ on $u$ and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfrak a}(u,\nabla\phi(u)) =\tilde{\mathfrak{f}}(b(u),\psi(u),\phi(u))+k(u)\mathfrak{a}_0(\nabla\phi(u)),$ with $\phi$ which is strictly increasing except on a locally finite number of segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If $b=\mathrm{Id}$, we obtain a general continuous dependence result on data $u_0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+\psi](\R)=\R$ and $\phi\circ [b+\psi]^{-1}$ is continuous