2018/12/05 by Maurizio Falcone, Falcone, Maurizio, Giulio Paolucci +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Biology Tumor Growth #Numerical Analysis (math.NA) #Numerical methods for differential equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1812.02140
openalex publication_date 2018/12/05 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We consider a class of "filtered" schemes for first order time dependent\nHamilton-Jacobi equations and prove a general convergence result for this class\nof schemes. A typical filtered scheme is obtained mixing a high-order scheme\nand a monotone scheme according to a filter function F which decides where\nthe scheme has to switch from one scheme to the other. A crucial role for this\nswitch is played by a parameter \ε=\ε(\Δ t,\Δ\nx)>0 which goes to 0 as the time and space steps (\Δ t,\Δ x) are\ngoing to 0 and does not depend on the time tn, for each iteration n. The\ntuning of this parameter in the code is rather delicate and has an influence on\nthe global accuracy of the filtered scheme. Here we introduce an adaptive and\nautomatic choice of \ε=\ε n (\Δ t, \Δ x) at every\niteration modifying the classical set up. The adaptivity is controlled by a\nsmoothness indicator which selects the regions where we modify the regularity\nthreshold \εn. A convergence result and some error estimates for\nthe new adaptive filtered scheme are proved, this analysis relies on the\nproperties of the scheme and of the smoothness indicators. Finally, we present\nsome numerical tests to compare the adaptive filtered scheme with other\nmethods.\n