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The Crank‐Nicolson/interpolating stabilized element‐free Galerkin method to investigate the fractional Galilei invariant advection‐diffusion equation
Author(s) -
Abbaszadeh Mostafa,
Dehghan Mehdi
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5871
Subject(s) - mathematics , crank–nicolson method , mathematical analysis , partial differential equation , fractional calculus , invariant (physics) , advection , galerkin method , convergence (economics) , numerical analysis , finite element method , mathematical physics , physics , economics , economic growth , thermodynamics
Recently, finding a stable and convergent numerical procedure to simulate the fractional partial differential equations (PDEs) is one of the interesting topics. Meanwhile, the fractional advection‐diffusion equation is a challenge model numerically and analytically. This paper develops a new meshless numerical procedure to simulate the fractional Galilei invariant advection‐diffusion equation. The fractional derivative is the Riemann‐Liouville fractional derivative sense. At the first stage, a difference scheme with the second‐order accuracy has been employed to get a semi‐discrete plan. After this procedure, the unconditional stability has been investigated, analytically. At the second stage, a meshless weak form based upon the interpolating stabilized element‐free Galerkin (ISEFG) method has been used to achieve a full‐discrete scheme. As for the full‐discrete scheme, the order of convergence is O ( τ 2 + r m + 1 ) . Two examples are studied, and simulation results are reported to verify the theoretical results.