scholarly journals All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations

CALCOLO ◽  
2021 ◽  
Vol 58 (4) ◽  
Author(s):  
Marco Donatelli ◽  
Rolf Krause ◽  
Mariarosa Mazza ◽  
Ken Trotti
Keyword(s):  
Diffusion Coefficients ◽  
Dimensional Space ◽  
Uniform Space ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Iterative Solvers ◽  
One Dimensional ◽  
Good Convergence ◽  
Constant Diffusion

AbstractWe focus on a time-dependent one-dimensional space-fractional diffusion equation with constant diffusion coefficients. An all-at-once rephrasing of the discretized problem, obtained by considering the time as an additional dimension, yields a large block linear system and paves the way for parallelization. In particular, in case of uniform space–time meshes, the coefficient matrix shows a two-level Toeplitz structure, and such structure can be leveraged to build ad-hoc iterative solvers that aim at ensuring an overall computational cost independent of time. In this direction, we study the behavior of certain multigrid strategies with both semi- and full-coarsening that properly take into account the sources of anisotropy of the problem caused by the grid choice and the diffusion coefficients. The performances of the aforementioned multigrid methods reveal sensitive to the choice of the time discretization scheme. Many tests show that Crank–Nicolson prevents the multigrid to yield good convergence results, while second-order backward-difference scheme is shown to be unconditionally stable and that it allows good convergence under certain conditions on the grid and the diffusion coefficients. The effectiveness of our proposal is numerically confirmed in the case of variable coefficients too and a two-dimensional example is given.

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

2020 ◽  
pp. 2150006
Author(s):  
Pratibha Verma ◽  
Manoj Kumar
Keyword(s):  
Analytical Solution ◽  
Diffusion Coefficients ◽  
High Efficiency ◽  
Dimensional Space ◽  
Adomian Decomposition Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Adomian Decomposition

In this paper, we have considered the multi-dimensional space fractional diffusion equations with variable coefficients. The fractional operators (derivative/integral) are used based on the Caputo definition. This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method (TSADM). Moreover, new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem. We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions. Finally, the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods. The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM. There are no restrictions imposed on the problems for diffusion coefficients, and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

scholarly journals A new method for solving a class of heat conduction equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1205-1210
Author(s):  
Yi Tian ◽  
Zai-Zai Yan ◽  
Zhi-Min Hong
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Dimensional Space ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Governing Equations ◽  
One Dimensional ◽  
Sparse System ◽  
Linear Algebraic Equations ◽  
Crank Nicolson

A numerical method for solving a class of heat conduction equations with variable coefficients in one dimensional space is demonstrated. This method combines the Crank-Nicolson and Monte Carlo methods. Using Crank-Nicolson method, the governing equations are discretized into a large sparse system of linear algebraic equations, which are solved by Monte Carlo method. To illustrate the usefulness of this technique, we apply it to two problems. Numerical results show the performance of the present work.

3D forward modeling of magnetotelluric fields in general anisotropic media and its numerical implementation in Julia

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. F29-F40 ◽  
Author(s):  
Bo Han ◽  
Yuguo Li ◽  
Gang Li
Keyword(s):  
Computational Cost ◽  
Forward Modeling ◽  
Iterative Solvers ◽  
Electrical Anisotropy ◽  
Staggered Grid ◽  
Good Convergence ◽  
Highly Efficient ◽  
Wide Range ◽  
Pure Matrix ◽  
Programming Process

We have developed a finite volume (FV) algorithm for magnetotelluric (MT) forward modeling in 3D conductivity structures with general anisotropy. The electric and magnetic fields are discretized on a conventional staggered grid, which cannot directly address the full-tensor conductivity. To overcome this difficulty, an interpolation scheme is used to average different components of the electric field to the same position. We formulate the algorithm in pure matrix form and implement it in a new language, Julia, making the programming process highly efficient and leading to a code with excellent readability, maintainability, and extendability. The validity of the FV Julia code is demonstrated using a layered 1D anisotropic model. For this model, the FV code provides accurate results, and the computational cost is reasonable. Being preconditioned with the electromagnetic potential ([Formula: see text]) system, the iterative solvers including quasi-minimal residual and biconjugate gradient stabilized exhibit a good convergence rate for a wide range of periods. The direct solvers MUMPS and PARDISO are highly efficient for small model sizes. For a relatively large model size with 2.18 millions unknowns, the linear system of one period can be solved by MUMPS within 360 s with multiple threads involved in the computation, and the memory usage is only 11.6 GB in the “out-of-core” mode. We further calculated MT responses of a 3D model with dipping and horizontal anisotropy, respectively. The results suggest that the electrical anisotropy can have significant influence on the MT response.

scholarly journals The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

2020 ◽  
pp. 875-889
Author(s):  
Firas A. Al-Saadawi ◽  
Hameeda Oda Al-Humedi
Keyword(s):  
Legendre Polynomials ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Bioheat Equation ◽  
One Dimensional ◽  
Good Convergence ◽  
The Matrix ◽  
The One ◽  
Fractional Type

The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.

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