A Compact Difference Scheme for Fourth-Order Temporal Multi-Term Fractional Wave Equations and Maximum Error Estimates

Liu, Guang-Hua Gao Rui

doi:10.4208/eajam.171118.060119

Cited by 7 publications

(3 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , u n-1 are existing and unique, we can obtain nonhomogeneous linear equations about u n from (13) and (15). If homogeneous linear equations only have a zero solution can be proved, we can prove that the existence and uniqueness of u n can be proved.…”

Section: Theoremmentioning

confidence: 99%

“…Cui structured a compact difference scheme for time fractional fourth-order equation with first Dirichlet boundary condition and discussed the stability and convergence [9]. Gao and Liu proposed a compact difference scheme for fourth-order temporal multi-term fractional wave equations and obtained maximum error estimates [13]. Later, Ji and Sun provided a high-order compact finite difference scheme for the fractional subdiffusion equation [16].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

Jiang

2020

Adv Differ Equ

View full text Add to dashboard Cite

Based on an L1 interpolation operator, a new high-order compact finite volume scheme is derived for the 2D multi-term time fractional sub-diffusion equation. It is shown that the difference scheme is unconditionally convergent and stable in $L_{\infty }$ L ∞ -norm. The convergence order is $O(\tau ^{2-\alpha }+h_{1}^{4}+h_{2}^{4})$ O ( τ 2 − α + h 1 4 + h 2 4 ) , where τ is the temporal step size and $h_{1}$ h 1 is the spatial step size in one direction, $h_{2}$ h 2 is the spatial step size in another direction. Two numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

show abstract

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

Jiang

2020

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…Since multi-term TFPDEs describe certain diffusion processes more accurately, various numerical methods, including Galerkin FEMs [3,17,50,56], orthogonal spline collocation methods [45], finite difference methods [5,16], compact difference methods [28], spectral methods [47,55], finite volume methods [46] have been developed for such equations. In addition, FEMs [35,36], compact finite difference methods [6,11], finite difference methods [38], Galerkin spectral element methods [6,30], singular boundary methods jointed with dual reciprocity methods [29] are also employed to multi-term TFWEs.…”

Section: Introductionmentioning

confidence: 99%

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

Fan¹

2021

EAJAM

View full text Add to dashboard Cite

An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical L1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable u and the flux p = µ(x)∇u, respectively, in H 1-and L 2-norms is derived by using the relationship between the projection operator R h and the interpolation operator I h. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

show abstract

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

Gao

Huang

Sun

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.

show abstract

A Compact Difference Scheme for Fourth-Order Temporal Multi-Term Fractional Wave Equations and Maximum Error Estimates

Cited by 7 publications

References 35 publications

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

Contact Info

Product

Resources

About