Finite Volume Hermite WENO Schemes for Solving the Hamilton-Jacobi Equation

2014 ◽  
Vol 15 (4) ◽  
pp. 959-980 ◽  
Author(s):  
Jun Zhu ◽  
Jianxian Qiu
Keyword(s):  
Finite Volume ◽  
Jacobi Equation ◽  
Two Dimensions ◽  
One Dimension ◽  
Weno Schemes ◽  
Numerical Tests ◽  
New Type ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

AbstractIn this paper, we present a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on the finite volume framework. The cell averages of the function and its first one (in one dimension) or two (in two dimensions) derivative values are together evolved via time approaching and used in the reconstructions. And the major advantages of the new HWENO schemes are their compactness in the spacial field, purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions. Extensive numerical tests are performed to illustrate the capability of the methodologies.

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

2019 ◽  
Vol 30 (02n03) ◽  
pp. 1950020 ◽  
Author(s):  
Xiaohan Cheng ◽  
Jianhu Feng ◽  
Supei Zheng ◽  
Xueli Song
Keyword(s):  
Finite Difference ◽  
Weno Schemes ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
New Type ◽  
Spurious Oscillations ◽  
Symmetry Requirement ◽  
Finite Difference Weno Schemes ◽  
Jacobi Equations ◽  
Fifth Order

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

scholarly journals Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

2021 ◽  
Vol 121 (2) ◽  
pp. 171-194
Author(s):  
Son N.T. Tu
Keyword(s):  
Rate Of Convergence ◽  
Classical Mechanics ◽  
One Dimension ◽  
Separable Potential ◽  
Periodic Homogenization ◽  
Periodic Functions ◽  
Effective Equation ◽  
Optimal Rate Of Convergence ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

scholarly journals Particle dynamics inside shocks in Hamilton–Jacobi equations

2010 ◽  
Vol 368 (1916) ◽  
pp. 1579-1593 ◽  
Author(s):  
Konstantin Khanin ◽  
Andrei Sobolevski
Keyword(s):  
Velocity Field ◽  
Jacobi Equation ◽  
Particle Dynamics ◽  
Velocity Fields ◽  
Vanishing Viscosity ◽  
Effective Velocity ◽  
Lagrangian Action ◽  
The Moment ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

scholarly journals Penetration into potential barriers in several dimensions

1937 ◽  
Vol 163 (915) ◽  
pp. 606-610 ◽  
Keyword(s):  
Refractive Index ◽  
Jacobi Equation ◽  
One Dimension ◽  
Practical Application ◽  
One Dimensional ◽  
Potential Barriers ◽  
Separable Problems ◽  
Total Reflexion ◽  
Similar Approximation ◽  
Hamilton Jacobi Equation

It is well known that in regions in which the refractive index varies sufficiently slowly, Schrödinger’s equation can be very simply treated by using its connexion with Hamilton-Jacobi’s differential equation. It is also known that a similar approximation is possible in regions of slowly varying imaginary refractive index (total reflexion). For the latter case the method was developed in papers by Jeffreys (1924), Wentzel (1926), Brillouin (1926) and Kramers (1926). These papers discuss also the behaviour of the wave function in the neighbourhood of the limit between the regions of real and imaginary refractive index. But although the connexion with the Hamilton-Jacobi equation holds in any number of dimensions, this equation can be solved by elementary means only in one dimension (or for problems that can by separation be reduced to one dimension), and for this reason the practical application of the method has so far been limited to one-dimensional or separable problems. In the present paper we discuss the case of more than one dimension and show that certain very simple inequalities may be obtained.

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

1963 ◽  
Vol 6 (3) ◽  
pp. 341-350 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Classical Particle ◽  
Quantum Mechanical ◽  
Order Partial Differential Equation ◽  
First Order ◽  
Relativistic Charged Particle ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

scholarly journals Instability of the Dirichlet problem for Hamilton-Jacobi equation

1997 ◽  
Vol 55 (2) ◽  
pp. 311-319
Author(s):  
Kewei Zhang
Keyword(s):  
Dirichlet Problem ◽  
Jacobi Equation ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation ◽  
General Conditions

We show the instability of solutions of the Dirichlet problem for Hamilton-Jacobi equations under quite general conditions.

A METRIC APPROACH TO PLASTICITY VIA HAMILTON–JACOBI EQUATION

2010 ◽  
Vol 20 (09) ◽  
pp. 1617-1647
Author(s):  
FERDINANDO AURICCHIO ◽  
ELENA BONETTI ◽  
ANTONIO MARIGONDA
Keyword(s):  
Jacobi Equation ◽  
Constitutive Relations ◽  
Yield Function ◽  
Generalized Gradients ◽  
Jacobi Equations ◽  
Thermodynamical Consistency ◽  
Hamilton Jacobi Equation ◽  
New Framework

Thermodynamical consistency of plasticity models is usually written in terms of the so-called "maximum dissipation principle". In this paper, we discuss constitutive relations for dissipative materials written through suitable generalized gradients of a (possibly non-convex) metric. This new framework allows us to generalize the classical results providing an interpretation of the yield function in terms of Hamilton–Jacobi equations theory.

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