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Riemann problem
About: Riemann problem is a research topic. Over the lifetime, 4331 publications have been published within this topic receiving 133669 citations.
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TL;DR: In this article, it is shown that these features can be obtained by constructing a matrix with a certain property U, i.e., property U is a property of the solution of the Riemann problem.
8,174 citations
Book•
01 Jan 1990
TL;DR: In this paper, the authors describe the derivation of conservation laws and apply them to linear systems, including the linear advection equation, the Euler equation, and the Riemann problem.
Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches
3,827 citations
TL;DR: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes.
Abstract: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.
3,133 citations
TL;DR: An hierarchy of uniformly high-order accurate schemes is presented which generalizes Godunov's scheme and its second- order accurate MUSCL extension to an arbitrary order of accuracy.
2,891 citations
Book•
27 Aug 1999
TL;DR: In this article, the generalized Riemann problem is used to solve the Euler Equation problem and the ADER approach is used for non-linear systems with finite forces in multiple dimensions.
Abstract: The Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014 linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014 State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013 Order and TVD Methods for Scalar Equations.- High#x2013 Order and TVD Schemes for Non#x2013 Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013 Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.
2,811 citations