Abstract
We present an abstract convergence result for the fixed point approximation of stationary Hamilton--Jacobi equations.The basic assumptions on the discrete operator are invariance with respect to the addition of constants, $\epsilon$-monotonicity and consistency.The result can be applied to various high-order approximation schemes which are illustrated in the paper.Several applications to Hamilton--Jacobi equations and numerical tests are presented.
Highlights
The numerical approximation of Hamilton-Jacobi equations ( HJ) plays a crucial role in many fields of application including optimal control, image processing, fluid dynamics, robotics and geophysics
The theory of approximation schemes for viscosity solutions has been developed starting from the huge literature existing for the numerical solution of conservation laws in one dimension
We prove an abstract convergence result for high-order methods relaxing the monotonicity assumption to ε-montonicity and show how some known schemes fit into this theory
Summary
The numerical approximation of Hamilton-Jacobi equations ( HJ) plays a crucial role in many fields of application including optimal control, image processing, fluid dynamics, robotics and geophysics This has motivated a number of different contributions where the main effort has been concentrated on the construction of schemes in multidimensional domains and on the conditions ensuring convergence to the weak solution (to be understood in this framework as the unique viscosity solution). In order to pass from a scheme for conservation laws to a scheme for the Hamilton-Jacobi equation one has to integrate in space the original scheme This approach is valid only in one dimension but, in practice, it has been extended to multidimensional problems using a dimensional splitting ([39, 42], see [38])
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