An efficient numerical method for the solution of 2D variable order time fractional mobile–immobile advection–dispersion model

2021 ◽  
Vol 44 (7) ◽  
pp. 5908-5929
Author(s):  
Marziyeh Saffarian ◽  
Akbar Mohebbi
Keyword(s):  
Numerical Method ◽  
Dispersion Model ◽  
Variable Order ◽  
Advection Dispersion

Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model

2016 ◽  
Vol 6 (3) ◽  
pp. 337-352 ◽  
Author(s):  
Heping Ma ◽  
Yubo Yang
Keyword(s):  
Fractional Derivative ◽  
Solute Transport ◽  
Collocation Method ◽  
Dispersion Model ◽  
Time Variable ◽  
Spectral Collocation Method ◽  
Variable Order ◽  
Spectral Collocation ◽  
Advection Dispersion ◽  
Variable Order Fractional Derivative

AbstractAn efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

Solving two-point seismic-ray tracing problems in a heterogeneous medium

1980 ◽  
Vol 70 (1) ◽  
pp. 79-99 ◽  
Author(s):  
V. Pereyra ◽  
W. H. K. Lee ◽  
H. B. Keller
Keyword(s):  
Numerical Method ◽  
Ray Tracing ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Nonlinear Boundary Conditions ◽  
Variable Order ◽  
Flexible Tool ◽  
Velocity Models ◽  
Jump Discontinuities ◽  
Variable Mesh

abstract A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

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