# Convergence of Adaptive Filtered Schemes for First Order Evolutionary Hamilton-Jacobi Equations

## TL;DR

This paper introduces an adaptive, automatic method for choosing the switching parameter in filtered schemes for first order Hamilton-Jacobi equations, improving accuracy and robustness through smoothness indicators and convergence analysis.

## Contribution

It proposes a novel adaptive and automatic approach for selecting the filter parameter in filtered schemes, enhancing convergence and accuracy for Hamilton-Jacobi equations.

## Key findings

- The adaptive scheme converges with proven error estimates.
- Numerical tests show improved accuracy over non-adaptive methods.
- The method effectively detects smoothness regions for optimal switching.

## Abstract

We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function $F$ which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter $\varepsilon=\varepsilon({\Delta t,\Delta x})>0$ which goes to 0 as the time and space steps $(\Delta t,\Delta x)$ are going to 0 and does not depend on the time $t_n$, for each iteration $n$. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of $\varepsilon=\varepsilon ^n (\Delta t, \Delta x)$ at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold $\varepsilon^n$. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.