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Efficient Implementation of Weighted ENO Schemes
, 1995
"... 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 [9]. 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 accur ..."
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Cited by 412 (38 self)
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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 [9]. 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 5th order WENO scheme for the case r = 3, instead of the 4th 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 4th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest to compute 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 donot 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 waveinteraction 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.
Non-oscillatory central differencing for hyperbolic conservation laws
- J. COMPUT. PHYS
, 1990
"... Many of the recently developed high-resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in orde ..."
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Cited by 298 (25 self)
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Many of the recently developed high-resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the “direction of the wind. ” Instead, we propose to use as a building block the more robust Lax-Friedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of the results obtained by such convenient central differencing is comparable with those of the upwind schemes.
Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 270 (26 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations
- J. Comput. Phys
, 2000
"... this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their high-resolution ..."
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Cited by 226 (22 self)
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this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their high-resolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semi-discrete formulation
A ‘‘vertically Lagrangian’’ finite-volume dynamical core for global models
- Weather Rev
, 2004
"... A finite-volume dynamical core with a terrain-following Lagrangian control-volume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow wate ..."
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Cited by 164 (10 self)
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A finite-volume dynamical core with a terrain-following Lagrangian control-volume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow water system. The 2D horizontalto-Lagrangian-surface transport and dynamical processes are then discretized using the genuinely conservative flux-form semi-Lagrangian algorithm. Time marching is split-explicit, with large time steps for scalar transport, and small fractional steps for the Lagrangian dynamics, which permits the accurate propagation of fast waves. A mass, momentum, and total energy conserving algorithm is developed for remapping the state variables periodically from the floating Lagrangian control-volume to an Eulerian terrain-following coordinate for dealing with ‘‘physical parameterizations’ ’ and to prevent severe distortion of the Lagrangian surfaces. Deterministic baroclinic wave-growth tests and long-term integrations using the Held–Suarez forcing are presented. Impact of the monotonicity constraint is discussed. 1.
An Implicit Upwind Algorithm for Computing Turbulent Flows on Unstructured Grids.
- Computers and Fluids
, 1994
"... Introduction For computing flows on complicated geometries such as multielement airfoils, the use of unstructured grids offers a good alternative to more traditional methods of analysis. This is primarily due to the promise of dramatically decreased time required to generate grids over complicated ..."
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Cited by 134 (17 self)
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Introduction For computing flows on complicated geometries such as multielement airfoils, the use of unstructured grids offers a good alternative to more traditional methods of analysis. This is primarily due to the promise of dramatically decreased time required to generate grids over complicated geometries. Also, unstructured grids offer the capability to locally adapt the grid to improve the accuracy of the computation without incurring the penalties associated with global refinement. While work remains to be done to fully realize their potential, much progress has been made in computing viscous flows on unstructured grids. While several advances have been made for computing turbulent flow on unstructured grids (e.g. [1] [2]), probably the most mature and widely used code for computing two-dimensional turbulent viscous flow on unstructured grids is that of Mavriplis [3]. In this reference, the solution algorithm is a Galerkin finite-element discretization and a Runge-Kutta
Nonoscillatory Central Schemes For Multidimensional Hyperbolic Conservation Laws
- SIAM J. Sci. Comput
, 1998
"... We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory ..."
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Cited by 133 (11 self)
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We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the com...
Analysis and Design of Numerical Schemes for Gas Dynamics 1 Artificial Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence
- INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS
, 1995
"... The theory of non-oscillatory scalar schemes is developed in this paper in terms of the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multi-dimensional problems on both structured and unstructured meshes, whi ..."
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Cited by 122 (45 self)
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The theory of non-oscillatory scalar schemes is developed in this paper in terms of the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multi-dimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for one-dimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multi-dimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with a modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.
