On weak and strong solutions for the neutron transport equation

Reed, Kennard W.

doi:10.1007/bf02788664

Cited by 113 publications

(127 citation statements)

References 9 publications

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“…Furthermore, the MPM space discretization leads to a decrease of the Courant number that prevents the capture of discontinuities. The discontinuous galerkin finite element method (DGFEM) has been developed for the neutron transport equation [2] and is now used in fluid and solid mechanics [3]. This extension of classical finite element method is based on the discontinuous Galerkin (DG) approximation in which shape functions are discontinuous across elements boundaries.…”

Section: Introductionmentioning

confidence: 99%

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. The material point method is extended in this work to the Discontinuous Galerkin approximation framework for the simulation of impacts on elastic and hyperelastic solids. The formulation is based on the weak form of conservation laws on each cell of an eulerian grid in which volume integrals are discretized on a set of material points lying in that cell, and on the computation of Godunov fluxes at cells faces. The resulting method is first derived within the small strains framework and illustrated on a one-dimensional and a two-dimensional problem of impact on an elastic media. Then a one-dimensional hyperelastic problem of a solid undergoing large strains is presented, and a comparison is performed with an analytical solution.

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Section: Introductionmentioning

confidence: 99%

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…Those types of schemes are referred to as Discontinuous Galerkin methods. They were proposed amongst others by REED and HILL in [11] with first analyses to be found in [6] and [4]. A…”

Section: Time-discontinuous Galerkin Methodsmentioning

confidence: 99%

Discontinuous Galerkin Methods for High-Dimensional Fokker-Planck Equations in Stochastic Dynamics

Loerke¹,

Nackenhorst²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…(iii) On the other extreme, the class of discontinuous Galerkin methods uses basis functions which are allowed to experience jump discontinuities across the interfaces [173,6]. These are particularly effective basis functions in problems with low regularity, such as the Eikonal equation [221] or problems with shock discontinuities [47,142].…”

Section: Methodsmentioning

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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On weak and strong solutions for the neutron transport equation

Cited by 113 publications

References 9 publications

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Discontinuous Galerkin Methods for High-Dimensional Fokker-Planck Equations in Stochastic Dynamics

A review of numerical methods for nonlinear partial differential equations

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