Björn Engquist

Some results on uniformly high-order accurate essentially nonoscillatory schemes

We continue the construction and the analysis of essentially nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy in the sense of truncation error, at extrema of the solution. In this paper we construct an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory. This means that, for piecewise smooth solutions, the variation of the numerical approximation is bounded by that of the true solution up to O(h^R^ ^-^ ^1), for 0

Triangle based adaptive stencils for the solution of hyperbolic conservation laws

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers equation are presented.

Absorbing boundary conditions for wave-equation migration

The standard boundary conditions used at the sides of a seismic section in wave-equation migration generate nartificial reflections. These reflections from the edges of the computational grid appear as artifacts in the final nsection. Padding the section with zero traces on either side adds to the cost of migration and simply delays the ninevitable reflections. nWe develop stable absorbing boundary conditions that annihilate almost all of the artificial reflections. This nis demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented can nbe used with any of the different types of finite-difference wave-equation migration, at essentially no extra cost.

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtls equation blow up in nite time

The Convergence Rate of Finite Difference Schemes in the Presence of Shocks

Finite difference approximations generically have

Absorbing boundary conditions for domain decomposition

{cal O}(1)

Numerical radiation boundary conditions for unsteady transonic flow

pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.

Wavelet-Based Numerical Homogenization with Applications

In this paper we would like to point out some similarities between two artificial boundary conditions. One is the far field or absorbing boundary conditions for computations over unbounded domain. The other is the boundary conditions used at the boundary between subdomains in domain decomposition. We show some convergence result for the generalized Schwarz alternating method (GSAM), in which a convex combination of Dirichlet data and Neumann data is exchanged at the artificial boundary. We can see clearly how the mixed boundary condition and the relative size of overlap will affect the convergence rate. These results can be extended to more general coercive elliptic partial differential equations using the equivalence of elliptic operators. Numerically first- and second-order approximations of the Dirichlet-to-Neumann operator are constructed using local operators, where information tangential to the boundary is included. Some other possible extensions and applications are pointed out. Finally numerical results are presented.

Far field boundary conditions for computation over long time

A family of numerical boundary conditions for far-field-computational boundaries in calculations involving unsteady transonic flow is devised. These boundary conditions are developed in a systematic fashion from general principles. Both numerical and analytic comparisons with other currently used methods are given.

Convergence of a Multigrid Method for Elliptic Equations with Highly Oscillatory Coefficients

Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.