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Conservation law
About: Conservation law is a research topic. Over the lifetime, 17340 publications have been published within this topic receiving 493018 citations.
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TL;DR: Fromm's second-order scheme for integrating the linear convection equation is made monotonic through the inclusion of nonlinear feedback terms in this paper, where care is taken to keep the scheme in conservation form.
2,005 citations
01 Jan 1998
TL;DR: In this paper, the authors describe the construction, analysis, and application of ENO and WENO schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations, where 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.
Abstract: 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 finite difference 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.
2,005 citations
Book•
01 Jan 1991
TL;DR: In this article, the authors present a model of a Tracer Particle in a Fluid with Hard Core Exclusion (TPE) and a Brownian Particle with hard core exclusion.
Abstract: Scales.- Outline.- I Classical Particles.- 1. Dynamics.- 1.1 Newtonian Dynamics.- 1.2 Boundary Conditions.- 1.3 Dynamics of Infinitely Many Particles.- 2. States of Equilibrium and Local Equilibrium.- 2.1 Equilibrium Measures, Correlation Functions.- 2.2 The Infinite Volume Limit.- 2.3 Local Equilibrium States.- 2.4 Local Stationarity.- 2.5 The Static Continuum Limit.- 3. The Hydrodynamic Limit.- 3.1 Propagation of Local Equilibrium.- 3.2 Hydrodynamic Equations.- 3.3 The Hard Rod Fluid.- 3.4 Steady States.- 4. Low Density Limit: The Boltzmann Equation.- 4.1 Low Density (Boltzmann-Grad) Limit.- 4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories.- 4.3 Convergence of the Scaled Correlation Functions.- 4.4 The Boltzmann Hierarchy.- 4.5 Time Reversal.- 4.6 Law of Large Numbers, Local Poisson.- 4.7 The H-Function.- 4.8 Extensions.- 5. The Vlasov Equation.- 6. The Landau Equation.- 7. Time Correlations and Fluctuations.- 7.1 Fluctuation Fields.- 7.2 The Green-Kubo Formula.- 7.3 Transport for the Hard Rod Fluid.- 7.4 The Fluctuating Boltzmann Equation.- 7.5 The Fluctuating Vlasov Equation.- 8. Dynamics of a Tracer Particle.- 8.1 Brownian Particle in a Fluid.- 8.2 The Stationary Velocity Process.- 8.3 Brownian Motion (Hydrodynamic) Limit.- 8.4 Large Mass Limit.- 8.5 Weak Coupling Limit.- 8.6 Low Density Limit.- 8.7 Mean Field Limit.- 8.8 External Forces and the Einstein Relation.- 8.9 Self-Diffusion.- 8.10 Corrections to Markovian Limits.- 9. The Role of Probability, Irreversibility.- II Stochastic Lattice Gases.- 1. Lattice Gases with Hard Core Exclusion.- 1.1 Dynamics.- 1.2 Stochastic Reversibility.- 1.3 Invariant Measures, Ergodicity, Domains of Attraction.- 1.4 Driven Lattice Gases.- 1.5 Standard Models.- 2. Equilibrium Fluctuations.- 2.1 Density Correlations and Bulk Diffusion.- 2.2 The Green-Kubo Formula.- 2.3 Currents.- 2.4 The Gradient Condition.- 2.5 Linear Response, Conductivity.- 2.6 Steady State Transport.- 2.7 State of Minimal Entropy Production.- 2.8 Bounds on the Conductivity.- 2.9 The Field of Density Fluctuations.- 2.10 Scaling Limit for the Density Fluctuation Field (Proof).- 2.11 Critical Dynamics.- 3. Nonequilibrium Dynamics for Reversible Lattice Gases.- 3.1 The Nonlinear Diffusion Equation.- 3.2 Hydrodynamic Limit (Proof).- 3.3 Low Temperatures.- 3.4 Weakly Driven Lattice Gases.- 3.5 Nonequilibrium Fluctuations.- 3.6 Local Equilibrium States and Minimal Entropy Production.- 3.7 Large Deviations.- 4. Nonequilibrium Dynamics of Driven Lattice Gases.- 4.1 Hyperbolic Equation of Conservation Type.- 4.2 Asymmetric Exclusion Dynamics.- 4.3 Fluctuation Theory.- 5. Beyond the Hydrodynamic Time Scale.- 5.1 Navier-Stokes Correction for Driven Lattice Gases.- 5.2 Local Structure of a Shock.- 5.2.1 Macroscopic Equation with Fluctuations.- 5.2.2 Shock in a Random Frame of Reference.- 5.2.3 Shock in Higher Dimensions.- 6. Tracer Dynamics.- 6.1 Two Component Systems.- 6.2 Tracer Diffusion.- 6.3 Convergence to Brownian Motion.- 6.4 Nearest Neighbor Jumps in One Dimension: The Case of Vanishing Self-Diffusion.- 7. Stochastic Models with a Single Conservation Law Other than Lattice Gases.- 7.1 Lattice Gases Without Hard Core/Zero Range Dynamics.- 7.2 Interacting Brownian Particles.- 7.3 Ginzburg-Landau Dynamics.- 8. Non-Hydrodynamic Limit Dynamics.- 8.1 Kinetic Limit.- 8.2 Mean Field Limit.- References.- List of Mathematical Symbols.
1,946 citations
Book•
01 Jan 1987
TL;DR: Quasi-linear Hyperbolic Equations Conservation Laws Single Conservation Laws The Decay of Solutions as t Tends to infinity Hypothesis of conservation laws Pairs of Conservation Laws as mentioned in this paper.
Abstract: Quasi-linear Hyperbolic Equations Conservation Laws Single Conservation Laws The Decay of Solutions as t Tends to Infinity Hyperbolic Systems of Conservation Laws Pairs of Conservation Laws Notes References.
1,919 citations
TL;DR: In this paper, the Runge?Kutta discontinuous Galerkin method for numerically solving hyperbolic conservation laws is extended to multidimensional nonlinear systems of conservation laws.
1,860 citations