Numerical methods with fourth-order spatial accuracy for variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid

Abstract

a r t i c l e i n f o Keywords:The variable-order nonlinear Stokes' first problem Heated generalized second grade fluid Stability Convergence Fourier analysis Fourth-order spatial accuracy Improved numerical method a b s t r a c t Stokes' first problem has in recent years received much attention. In this paper, we focus on the variable-order nonlinear Stokes' first problem for a heated generalized second grade fluid. A numerical scheme with fourth-order spatial accuracy is developed to solve the problem. The stability, solvability and convergence of the numerical scheme are discussed via Fourier analysis. An improved numerical scheme is also developed. In addition, a numerical example is given and the numerical results support the effectiveness of our theoretical analysis results.

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

“…In this section, we will structure another numerical scheme for solving problem (1)-(4), which improves the accuracy of IFDS (13)- (15).…”

“…In this section, we use the Fourier method to discuss the stability of the IFDS (13)- (15). Let z k j denote the approximate solution of IFDS (13)- (15) and…”

“…In the same year, Wu [16] also gave a numerical scheme for solving the above mentioned problem and proved the scheme also has convergence order O(τ + h 2 ). Moreover, in 2011, Chen et al [15] considered the variable-order nonlinear Stokes' first problem for a heated generalized second grade fluid and obtained two numerical schemes, which have convergence order O(τ + h 4 ) and O(τ 2 + h 4 ), respectively.…”

“…For more numerical theoretical results, we refer to [1,5,[12][13][14][15] and the references cited therein.…”

Numerical studies for the variable-order nonlinear fractional wave equation

“…This method is used to solve many of similar problems to the proposed problem, for example, the variable-order nonlinear cable equation [1], the variable-order nonlinear Stokes' first problem for a heated generalized second grade fluid [2], the variable-order anomalous sub-diffusion equation [4], the space-time Riesz-Caputo fractional advection-diffusion equation [19], the fractional partial differential equations with Riesz space fractional derivatives [24], the variable-order fractional advection-diffusion with a nonlinear source term [25], and others ( [8]- [11], [20], [21]). Definition 1.1.…”