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High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation
Author(s) -
Bak Soyoon
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22374
Subject(s) - discretization , mathematics , nonlinear system , advection , temporal discretization , tracking (education) , convergence (economics) , diffusion , burgers' equation , rate of convergence , finite difference method , convection–diffusion equation , mathematical analysis , control theory (sociology) , mathematical optimization , partial differential equation , computer science , psychology , pedagogy , channel (broadcasting) , computer network , physics , control (management) , quantum mechanics , artificial intelligence , economics , thermodynamics , economic growth
In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L ‐stable and third‐order L (α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.