Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes, II

Shu, Chi‐Wang; Osher, Stanley

doi:10.1007/978-3-642-60543-7_14

Cited by 455 publications

(692 citation statements)

References 12 publications

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“…To retain high-order accuracy in time without creating spurious oscillations, it is customary to use a TVD Runge-Kutta method [17,18]. These Runge-Kutta methods involve a convex combination of forward Euler steps to advance the solution in time and are designed to ensure that the solution is total-variation diminishing.…”

Section: Positive Schemesmentioning

confidence: 99%

Positive Scheme Numerical Simulation of High Mach Number Astrophysical Jets

Gardner

2007

J Sci Comput

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High Mach number astrophysical jets are simulated using a positive scheme, and are compared with WENO-LF simulations. A version of the positive scheme has allowed us to simulate astrophysical jets with radiative cooling up to Mach number 270 with respect to the heavy jet gas, a factor of two times higher than the maximum Mach number attained with the WENO schemes and ten times higher than with CLAWPACK. Such high Mach numbers occur in many settings in astrophysical flows, so it is important to develop a scheme that can simulate at these Mach numbers.

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Section: Positive Schemesmentioning

confidence: 99%

Positive Scheme Numerical Simulation of High Mach Number Astrophysical Jets

Gardner

2007

J Sci Comput

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“…This test case proposed in [46], describes the interaction of a right moving shock with an oscillatory smooth wave. Its initial condition is given by The solution is simulated on a mesh with N = 256 cells, until the time T = 1.8 and CFL=0.1.…”

Section: Shock-entropy Testmentioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

120

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High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

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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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Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes, II

Cited by 455 publications

References 12 publications

Positive Scheme Numerical Simulation of High Mach Number Astrophysical Jets

Positive Scheme Numerical Simulation of High Mach Number Astrophysical Jets

An artificial neural network as a troubled-cell indicator

A review of numerical methods for nonlinear partial differential equations

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