A fast temporal second-order compact ADI difference scheme for the 2D multi-term fractional wave equation

2020 ◽  
Author(s):  
Hong Sun ◽  
Zhi-zhong Sun
Keyword(s):  
Wave Equation ◽  
Difference Scheme ◽  
Second Order ◽  
Fractional Wave Equation

scholarly journals Numerical Algorithms for the Fractional Diffusion-Wave Equation with Reaction Term

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Wave Equation ◽  
Difference Scheme ◽  
Fourier Method ◽  
Numerical Algorithms ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Wave ◽  
Order Difference ◽  
Reaction Term ◽  
Liouville Derivative

Two numerical algorithms are derived to compute the fractional diffusion-wave equation with a reaction term. Firstly, using the relations between Caputo and Riemann-Liouville derivatives, we get two equivalent forms of the original equation, where we approximate Riemann-Liouville derivative by a second-order difference scheme. Secondly, for second-order derivative in space dimension, we construct a fourth-order difference scheme in terms of the hyperbolic-trigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

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