In recent years, Hamilton-Jacobi equations have attracted a lot of attention due to the important role that they play in various fields of applications.In this book, we introduce recent developments in adaptive numerical methods for Hamilton-Jacobi equations.We construct convergent adaptive high order schemes for time dependent Hamilton-Jacobi equations and propose a general method of constructing convergent adapive high order schemes and discuss the question of its convergence.Then, we propose a method of constructing convergent adaptive high order schemes for Hamilton-Jacobi equations on triangular meshes. According to this methodology, we construct adaptive schemes of WENO type on triangular meshes.Finally, we develop WENO schemes by newly defining nonlinear weights during WENO reconstruction which are not only simpler but also have clearer meaning than previous ones. Based on the analysis of numerical properties of the improved WENO scheme, we propose an algorithm of constructing a convergent adaptive WENO scheme by adapting the proposed WENO scheme through a newly introduced singularity indicator.

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