Ami Harten

High Resolution Schemes for Hyperbolic Conservation Laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

Uniformly high order accurate essentially non-oscillatory schemes, 111

We continue the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws. We present an hierarchy of uniformly high-order accurate schemes which generalizes Godunovs scheme and its second-order accurate MUSCL extension to an arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and, consequently, the resulting schemes are highly nonlinear.

Uniformly high-order accurate nonoscillatory schemes

We begin the construction and the analysis of 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 a uniformly second-order approximation, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise-linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem and an average of this approximate solution over each cell.

On a Class of High Resolution Total-Variation-Stable Finite-Difference Schemes

This paper presents a class of explicit and implicit second order accurate finite-difference schemes for the computation of weak solutions of hyperbolic conservation laws. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux. The so derived second order accurate schemes achieve high resolution, while retaining the robustness of the original first order accurate scheme.

Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆

It is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws. The grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals. The resulting algorithm is a simple extension of many currently popular Godunov-type methods. Computer codes using one of these methods can be easily modified to add the moving mesh as an option. Numerical examples are given that illustrate the improved accuracy of Godunovs and Roes methods on a self-adjusting mesh.

ENO schemes with subcell resolution

In this paper, we introduce the notion of subcell resolution, which is based on the observation that unlike point values, cell-averages of a discontinuous piecewise-smooth function contain information about the exact location of the discontinuity within the cell. Using this observation we design an essentially non-oscillatory (ENO) reconstruction technique which is exact for cell averages of discontinuous piecewise-polynomial functions of the appropriate degree. Later on we incorporate this new reconstruction technique into Godunov-type schemes in order to produce a modification of the ENO schemes which prevents the smearing of contact discontinuities.

Multiresolution representation of data: a general framework

In this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings. This framework, which can be viewed as a generalization of the theory of wavelets, includes discretization corresponding to unstructured grids in several space dimensions, and thus is general enough to enable us to embed most numerical problems in a multiresolution setting. Furthermore, this framework allows for nonlinear (data-dependent) multiresolution representation schemes and thus enables us to design adaptive data-compression algorithms.In this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis.

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

Self-adjusting hybrid schemes for shock computations

Abstract A general approach for constructing “hybrid schemes” in order to solve quasilinear hyperbolic initial-value problems is presented. The hybrid schemes are constructed from a first-order accuracy operator which dominates in shock regions and a higher-order operator which produces more accurate results in smooth regions. The usefulness and accuracy of the method is demonstrated in one- and two-dimensional examples, while overcoming nonlinear instabilities and post-shock oscillations.

Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I

A generalization of Hartens multiresolution algorithms to two-dimensional (2-D) hyperbolic conservation laws is presented. Given a Cartesian grid and a discretized function on it, the method computes the local-scale components of the function by recursive diadic coarsening of the grid. Since the functions regularity can be described in terms of its scale or multiresolution analysis, the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth. Instead, in those locations, the divergence of the solution is interpolated from the next coarser grid level. First, the basic 2-D essentially nonoscillatory (ENO) scheme is presented, then the 2-D multiresolution analysis is developed, and finally the subsequent scheme is tested numerically. The computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well.