A difference scheme for a nonlinear partial integrodifferential equation
SIAM Journal on Numerical Analysis
A finite difference scheme for partial integro-differential equations with a weakly singular kernel
Applied Numerical Mathematics
Finite difference approximations for fractional advection-dispersion flow equations
Journal of Computational and Applied Mathematics
The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
A second-order accurate numerical approximation for the fractional diffusion equation
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
A Fourier method for the fractional diffusion equation describing sub-diffusion
Journal of Computational Physics
Numerical algorithm for the time fractional Fokker-Planck equation
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
A fully discrete difference scheme for a diffusion-wave system
Applied Numerical Mathematics
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
A compact finite difference scheme for the fractional sub-diffusion equations
Journal of Computational Physics
Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation
SIAM Journal on Scientific Computing
A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions
Journal of Computational Physics
Computers & Mathematics with Applications
Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation
Journal of Computational Physics
Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
Journal of Computational Physics
Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions
Journal of Computational Physics
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A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The $$L1$$ discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is $$\mathcal{O }(\tau ^{3-\alpha }+h^4)$$ in the maximum norm, where $$\tau $$ is the temporal grid size and $$h$$ is the spatial grid size, respectively. In addition, a Crank---Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of $$\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)$$ is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank---Nicolson scheme are presented to show the effectiveness of the compact scheme.