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Shu, Chi-Wang

Publications and source records attributed to Shu, Chi-Wang.

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NASA NTRS1995

Efficient implementation of weighted ENO schemes

In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher and Chan. It was shown by Liu et al. that WENO schemes constructed from the r-th order (in L1 norm) ENO schemes are (r+1)-th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5-th order WENO scheme for the case r = 3, instead of the 4-th order with the original smoothness measurement by Liu et al. This 5-th order WENO scheme is as fast as the 4-th order WENO scheme of Liu et al., and both schemes are about twice as fast as the 4-th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest computing the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which do not contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy wave interaction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.

Jiang, Guang-Shan↗
NASA NTRS1994

On the Gibbs phenomenon 5: Recovering exponential accuracy from collocation point values of a piecewise analytic function

The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct a uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.

Gottlieb, David↗
NASA NTRS1994

A note on the accuracy of spectral method applied to nonlinear conservation laws

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear partial differential equation with a discontinuous solution, Fourier spectral method produces poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials.

Shu, Chi-Wang↗
NASA NTRS1994

On the Gibbs phenomenon 4: Recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function

We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(sub k)(sup mu)(x) with the weight function (1 - x(exp 2))(exp mu - 1/2) for any constant mu is greater than or equal to 0, of an L(sub 1) function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

Gottlieb, David↗
NASA NTRS1994

A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow

High order essentially non-oscillatory (ENO) schemes, originally designed for compressible flow and in general for hyperbolic conservation laws, are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The projection to divergence-free velocity fields is achieved by fourth-order central differences through fast Fourier transforms (FFT) and a mild high-order filtering. The objective of this work is to assess the resolution of ENO schemes for large scale features of the flow when a coarse grid is used and small scale features of the flow, such as shears and roll-ups, are not fully resolved. It is found that high-order ENO schemes remain stable under such situations and quantities related to large scale features, such as the total circulation around the roll-up region, are adequately resolved.

Weinan, E.↗
NASA NTRS1993

On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.

Gottlieb, David↗
NASA NTRS1993

On positivity preserving finite volume schemes for compressible Euler equations

Positivity preserving property of first and higher order finite volume schemes for one and two dimensional compressible Euler equations of gas dynamics is considered. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve high order positivity preserving reconstruction is described.

Perthame, Benoit↗
NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially non-oscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, Jay↗
NASA NTRS1993

Effective equations and the inverse cascade theory for Kolmogorov flows

We study the two-dimensional Kolmogorov flows in the limit as the forcing frequency goes to infinity. Direct numerical simulation indicates that the low frequency energy spectrum evolves to a universal kappa (exp -4) decay law. We derive effective equations governing the behavior of the large scale flow quantities. We then present numerical evidence that with smooth initial data, the solution to the effective equation develops a kappa (exp -4) type singularity at a finite time. This gives a convenient explanation for the kappa (exp -4) decay law exhibited by the original Kolmogorov flows.

Weinan, E.↗
NASA NTRS1993

Uniform high-order spectral methods for one- and two-dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗
NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially nonoscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, J.↗
NASA NTRS1992

A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow

High order essentially non-oscillatory (ENO) schemes, originally designed for compressible flow and in general for hyperbolic conservation laws, are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The projection to divergence-free velocity fields is achieved by fourth order central differences through Fast Fourier Transforms (FFT) and a mild high-order filtering. The objective of this work is to assess the resolution of ENO schemes for large scale features of the flow when a coarse grid is used and small scale features of the flow, such as shears and roll-ups, are not fully resolved. It is found that high-order ENO schemes remain stable under such situations and quantities related to large-scale features, such as the total circulation around the roll-up region, are adequately resolved.

Weinan, E.↗
NASA NTRS1992

Numerical study of the small scale structures in Boussinesq convection

Two-dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high order accurate Essentially Nonoscillatory (ENO) scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. The numerical results suggest that the collapse of the bubble cap is unlikely to occur in resolved calculations. The strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. We also found that the cascade of energy to small scales is dominated by the formulation of thin and sharp fronts across which density jumps.

Weinan, E.↗
NASA NTRS1992

Multi-dimensional high order essentially non-oscillatory finite difference methods in generalized coordinates

The nonlinear stability of compact schemes for shock calculations is investigated. In recent years compact schemes were used in various numerical simulations including direct numerical simulation of turbulence. However to apply them to problems containing shocks, one has to resolve the problem of spurious numerical oscillation and nonlinear instability. A framework to apply nonlinear limiting to a local mean is introduced. The resulting scheme can be proven total variation (1D) or maximum norm (multi D) stable and produces nice numerical results in the test cases. The result is summarized in the preprint entitled 'Nonlinearly Stable Compact Schemes for Shock Calculations', which was submitted to SIAM Journal on Numerical Analysis. Research was continued on issues related to two and three dimensional essentially non-oscillatory (ENO) schemes. The main research topics include: parallel implementation of ENO schemes on Connection Machines; boundary conditions; shock interaction with hydrogen bubbles, a preparation for the full combustion simulation; and direct numerical simulation of compressible sheared turbulence.

Shu, Chi-Wang↗
NASA NTRS1992

Resolution properties of the Fourier method for discontinuous waves

In this paper we discuss the wave-resolution properties of the Fourier approximations of a wave function with discontinuities. It is well known that a minimum of two points per wave is needed to resolve a periodic wave function using Fourier expansions. For Chebyshev approximations of a wave function, a minimum of pi points per wave is needed. Here we obtain an estimate for the minimum number of points per wave to resolve a discontinuous wave based on its Fourier coefficients. In our recent work on overcoming the Gibbs phenomenon, we have shown that the Fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficient of a rapidly converging Gegenbauer expansion. We therefore study the resolution properties of a Gegenbauer expansion where both the number of terms and the order increase.

Gottlieb, David↗
NASA NTRS1992

A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting

The present treatment of elliptic regions via hyperbolic flux-splitting and high order methods proposes a flux splitting in which the corresponding Jacobians have real and positive/negative eigenvalues. While resembling the flux splitting used in hyperbolic systems, the present generalization of such splitting to elliptic regions allows the handling of mixed-type systems in a unified and heuristically stable fashion. The van der Waals fluid-dynamics equation is used. Convergence with good resolution to weak solutions for various Riemann problems are observed.

Shu, Chi-Wang↗
NASA NTRS1992

Nonlinearly stable compact schemes for shock calculations

The applications of high-order, compact finite difference methods in shock calculations are discussed. The main concern is to define a local mean which will serve as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while maintaining the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total-variation stable in one space dimension and maximum-norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. These ideas can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.

Cockburn, Bernardo↗