On discreteness of the Hopf equation

DOI10.1007/s10255-008-8021-1zbMath1149.65071OpenAlexW2104137044MaRDI QIDQ946238

Hai-liang Liu

Publication date: 22 September 2008

Published in: Acta Mathematicae Applicatae Sinica. English Series (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10255-008-8021-1

zbMATH Keywords

hyperbolic conservation lawslevel set equationHopf equationmulti-valued solutionscentered schemedispersive schemeGoudman-Lax schemeshock capturing schemes

Mathematics Subject Classification ID

Hyperbolic conservation laws (35L65) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06)

Related Items

Wave breaking and propagation speed for a class of nonlocal dispersive \(\theta \)-equations ⋮ On \(H^2\)-solutions for a Camassa-Holm type equation ⋮ On traveling wave solutions of the \(\theta\)-equation of dispersive type ⋮ Well-posedness and blow-up phenomena for a higher order shallow water equation ⋮ Bifurcation of traveling wave solutions for a two-component generalized \(\theta\)-equation ⋮ An Invariant Preserving Discontinuous Galerkin Method for the Camassa--Holm Equation ⋮ The Cauchy problem for a two-component generalized \(\theta \)-equations ⋮ Optimal control of a class of nonlocal dispersive equations ⋮ Optimal control of a viscous generalized \(\theta\)-type dispersive equation with weak dissipation ⋮ Abundant dynamical behaviors of bounded traveling wave solutions to generalized \(\theta\)-equation ⋮ Bifurcation of solitons, peakons, and periodic cusp waves for \(\theta\)-equation

Uses Software

• HLLE

Cites Work

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• Superposition of multi-valued solutions in high frequency wave dynamics
• Geometric optics in a phase-space-based level set and Eulerian framework
• The piecewise parabolic method (PPM) for gas-dynamical simulations
• On the well-posedness of the Degasperis-Procesi equation
• A level set framework for capturing multi-valued solutions of nonlinear first-order equations
• Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations
• Non-oscillatory central differencing for hyperbolic conservation laws
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• High resolution schemes for hyperbolic conservation laws
• Uniformly high order accurate essentially non-oscillatory schemes. III
• Efficient implementation of essentially nonoscillatory shock-capturing schemes
• Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
• Efficient implementation of essentially nonoscillatory shock-capturing schemes. II
• Third order nonoscillatory central scheme for hyperbolic conservation laws
• On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices
• Weighted essentially non-oscillatory schemes
• Convergence results for an inhomogeneous system arising in various high frequency approximations
• New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations
• Runge--Kutta discontinuous Galerkin methods for convection-dominated problems
• Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation
• Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
• On dispersive difference schemes
• Level set methods and dynamic implicit surfaces
• Global weak solutions for a new periodic integrable equation with peakon solutions
• Local level set method in high dimension and codimension
• Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids
• Modulation of multiphase waves in the presence of resonance
• Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics
• Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method
• A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations
• A level set-based Eulerian approach for anisotropic wave propagation
• Central schemes on overlapping cells
• Large oscillations arising in a dispersive numerical scheme
• Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems
• A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations
• Computational high-frequency wave propagation using the level set method with applications to the semi-classical limit of Schrödinger equations
• Strong Stability-Preserving High-Order Time Discretization Methods
• A LEVEL SET APPROACH TO SEMICONTINUOUS VISCOSITY SOLUTIONS FOR CAUCHY PROBLEMS
• Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations
• Critical Thresholds in a Convolution Model for Nonlinear Conservation Laws
• Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
• A Level Set Formulation for the Solution of the Dirichlet Problem for Hamilton–Jacobi Equations
• Two approximations of solutions of Hamilton-Jacobi equations
• The small dispersion limit of the Korteweg-de Vries equation. I
• Dispersive approximations in fluid dynamics
• Wave Breaking in a Class of Nonlocal Dispersive Wave Equations
• AN ALTERNATING EVOLUTION APPROXIMATION TO SYSTEMS OF HYPERBOLIC CONSERVATION LAWS
• A NONLOCAL CONSERVATION LAW WITH NONLINEAR "RADIATION" INHOMOGENEITY
• On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws
• The zero dispersion limit of the korteweg-de vries equation for initial potentials with non-trivial reflection coefficient
• Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes
• Riemann Solvers, the Entropy Condition, and Difference
• On dispersive difference schemes. I
• Central WENO schemes for hyperbolic systems of conservation laws
• Stable and Entropy Satisfying Approximations for Transonic Flow Calculations
• The toda shock problem
• Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws
• The Riemann problem for the transportation equations in gas dynamics
• The Small Dispersion Limit for a Nonlinear Semidiscrete System of Equations
• Formation of $\delta$-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids
• Computational high frequency wave propagation
• High-Order Central Schemes for Hyperbolic Systems of Conservation Laws
• Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness
• A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
• A Geometric Interpretation of the Heat Equation with Multivalued Initial Data
• Difference schemes for hyperbolic equations with high order of accuracy
• On the solution of nonlinear hyperbolic differential equations by finite differences
• Weak solutions of nonlinear hyperbolic equations and their numerical computation
• A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems