The Piecewise Parabolic Method (PPM) for gas-dynamical simulations

Colella, Phillip; Woodward, Paul R.

doi:10.1016/0021-9991(84)90143-8

Cited by 3,646 publications

(2,883 citation statements)

References 8 publications

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“…The answer to this question is problem dependent. For many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems (shock tube problems), a good second order method, such as PPM [4] or other TVD [6] methods, would be the optimal choice. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method may be more economical in CPU time, as demonstrated by the examples in this paper.…”

Section: Introductionmentioning

confidence: 99%

Resolution of high order WENO schemes for complicated flow structures

Shi

Zhang

Shu

2003

Journal of Computational Physics

267

122

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In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially nonoscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.

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Section: Introductionmentioning

confidence: 99%

Resolution of high order WENO schemes for complicated flow structures

Shi

Zhang

Shu

2003

Journal of Computational Physics

267

122

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“…Advective terms in POLCOMS are computed using the piecewise parabolic method (PPM, Colella and Woodward, 1984), which has low numerical diffusion and better preserves frontal features (James, 1996;1997). By default POLCOMS does not apply explicit horizontal diffusion and at the Liverpool Bay scale consideration of diffusion has little impact on the frontal position (Norman et al, 2014d).…”

Section: Modelling Approachmentioning

confidence: 99%

Impact of operational model nesting approaches and inherent errors for coastal simulations

et al. 2016

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A region of freshwater influence (ROFI) under hypertidal conditions is used to demonstrate inherent problems for nested operational modelling systems. Such problems can impact the accurate simulation of freshwater export within shelf seas, so must be considered in coastal ocean modelling studies. In Liverpool Bay (our UK study site), freshwater inflow from 3 large estuaries forms a coastal front that moves in response to tides and winds. The cyclic occurrence of stratification and remixing is important for the biogeochemical cycles, as nutrient and pollutant loaded freshwater is introduced into the coastal system. Validation methods, using coastal observations from fixed moorings and cruise transects, are used to assess the simulation of the ROFI, through improved spatial structure and temporal variability of the front, as guidance for best practise model setup. A structured modelling system using a 180 m grid nested within a 1.8 km grid demonstrates how compensation for error at the coarser resolution can have an adverse impact on the nested, high resolution application. Using 2008, a year of typical calm and stormy periods with variable river influence, the sensitivities of the ROFI dynamics to initial and boundary conditions are investigated. It is shown that accurate representation of the initial water column structure is important at the regional scale and that the boundary conditions are most important at the coastal scale. Although increased grid resolution captures the frontal structure, the accuracy in frontal position is determined by the offshore boundary conditions and therefore the accuracy of the coarser regional model.

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“…The intense period of development of such limiters during the 1980s and 1990s was marked by a series of acronyms such as the second-order MUSCL and TVD scheme [135,100,192], cubic-order PPM scheme [50], and the class of higher-order (W)ENO schemes [102,101,186,141,49,79].…”

Section: = 1 δTmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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The Piecewise Parabolic Method (PPM) for gas-dynamical simulations

Cited by 3,646 publications

References 8 publications

Resolution of high order WENO schemes for complicated flow structures

Resolution of high order WENO schemes for complicated flow structures

Impact of operational model nesting approaches and inherent errors for coastal simulations

A review of numerical methods for nonlinear partial differential equations

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