A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A RBF meshless approach for modeling a fractal mobile/immobile transport model

A fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBFs) to discretize the space variable. In contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example which is presented to describe a fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating fractional differential equations, and it has good potential in the development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Finite volume methods for simulating anomalous transport

In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems

In this article, we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction-diffusion problems having diffusion parameters with different magnitudes. The central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle. An a priori monitor function is obtained from the error estimate. A suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second-order parameter uniform convergence. We present the results of numerical experiments for linear and semilinear reaction-diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis. (C) 2014 Elsevier Inc. All rights reserved.

Diffusion analogy and shock capturing for solving aerodynamic equations

Improving the resolution of the shock is one of the most important subjects in computational aerodynamics. In this paper the behaviour of the solutions near the shock is discussed and the reason of the oscillation production is investigated heuristically. According to the differential approximation of the difference scheme the so-called diffusion analogy equation and the diffusion analogy coefficient are defined. Four methods for improving the resolution of the shock are presented using the concept of diffusion analogy.

SIMPLE REMESHING SCHEME FOR THE FINITE ELEMENT BASED SIMULATION OF DOPANT DIFFUSION IN SILICON.

Process simulation programs are valuable in generating accurate impurity profiles. Apart from accuracy the programs should also be efficient so as not to consume vast computer memory. This is especially true for devices and circuits of VLSI complexity. In this paper a remeshing scheme to make the finite element based solution of the non-linear diffusion equation more efficient is proposed. A remeshing scheme based on comparing the concentration values of adjacent node was then implemented and found to remove the problems of oscillation.

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

Maximum principle preserving high order schemes for convection-dominated diffusion equations

The maximum principle is an important property of solutions to PDE. Correspondingly, it's of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO scheme, and then propose a new parametrized MPP high order finite difference(FD) WENO framework, which is generalized from the one solving hyperbolic conservation laws. A formal analysis is presented to show that a third order finite difference scheme with this parametrized MPP flux limiters maintains the third order accuracy without extra CFL constraint when the low order monotone flux is chosen appropriately. Numerical tests in both one and two dimensional cases are performed on the simulation of the incompressible Navier-Stokes equations in vorticity stream-function formulation and several other problems to show the effectiveness of the proposed method.

Un método de diferencias finitas generalizadas para simular la actividad eléctrica de un tejido cardiaco

Una evolución del método de diferencias finitas ha sido el desarrollo del método de diferencias finitas generalizadas (MDFG) que se puede aplicar a mallas irregulares o nubes de puntos. En este método se emplea una expansión en serie de Taylor junto con una aproximación por mínimos cuadrados móviles (MCM). De ese modo, las fórmulas explícitas de diferencias para nubes irregulares de puntos se pueden obtener fácilmente usando el método de Cholesky. El MDFG-MCM es un método sin malla que emplea únicamente puntos. Una contribución de esta Tesis es la aplicación del MDFG-MCM al caso de la modelización de problemas anisótropos elípticos de conductividad eléctrica incluyendo el caso de tejidos reales cuando la dirección de las fibras no es fija, sino que varía a lo largo del tejido. En esta Tesis también se muestra la extensión del método de diferencias finitas generalizadas a la solución explícita de ecuaciones parabólicas anisótropas. El método explícito incluye la formulación de un límite de estabilidad para el caso de nubes irregulares de nodos que es fácilmente calculable. Además se presenta una nueva solución analítica para una ecuación parabólica anisótropa y el MDFG-MCM explícito se aplica al caso de problemas parabólicos anisótropos de conductividad eléctrica. La evidente dificultad de realizar mediciones directas en electrocardiología ha motivado un gran interés en la simulación numérica de modelos cardiacos. La contribución más importante de esta Tesis es la aplicación de un esquema explícito con el MDFG-MCM al caso de la modelización monodominio de problemas de conductividad eléctrica. En esta Tesis presentamos un algoritmo altamente eficiente, exacto y condicionalmente estable para resolver el modelo monodominio, que describe la actividad eléctrica del corazón. El modelo consiste en una ecuación en derivadas parciales parabólica anisótropa (EDP) que está acoplada con un sistema de ecuaciones diferenciales ordinarias (EDOs) que describen las reacciones electroquímicas en las células cardiacas. El sistema resultante es difícil de resolver numéricamente debido a su complejidad. Proponemos un método basado en una separación de operadores y un método sin malla para resolver la EDP junto a un método de Runge-Kutta para resolver el sistema de EDOs de la membrana y las corrientes iónicas. ABSTRACT An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method that can be applied to irregular grids or clouds of points. In this method a Taylor series expansion is used together with a moving least squares (MLS) approximation. Then, the explicit difference formulae for irregular clouds of points can be easily obtained using a simple Cholesky method. The MLS-GFD is a mesh-free method using only points. A contribution of this Thesis is the application of the MLS-GFDM to the case of modelling elliptic anisotropic electrical conductivity problems including the case of real tissues when the fiber direction is not fixed, but varies throughout the tissue. In this Thesis the extension of the generalized finite difference method to the explicit solution of parabolic anisotropic equations is also given. The explicit method includes a stability limit formulated for the case of irregular clouds of nodes that can be easily calculated. Also a new analytical solution for homogeneous parabolic anisotropic equation has been presented and an explicit MLS- GFDM has been applied to the case of parabolic anisotropic electrical conductivity problems. The obvious difficulty of performing direct measurements in electrocardiology has motivated wide interest in the numerical simulation of cardiac models. The main contribution of this Thesis is the application of an explicit scheme based in the MLS-GFDM to the case of modelling monodomain electrical conductivity problems using operator splitting including the case of anisotropic real tissues. In this Thesis we present a highly efficient, accurate and conditionally stable algorithm to solve a monodomain model, which describes the electrical activity in the heart. The model consists of a parabolic anisotropic partial differential equation (PDE), which is coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. The resulting system is challenging to solve numerically, because of its complexity. We propose a method based on operator splitting and a meshless method for solving the PDE together with a Runge-Kutta method for solving the system of ODE’s for the membrane and ionic currents.

Difference schemes for time-dependent heat conduction models with delay

Different non-Fourier models of heat conduction, that incorporate time lags in the heat flux and/or the temperature gradient, have been increasingly considered in the last years to model microscale heat transfer problems in engineering. Numerical schemes to obtain approximate solutions of constant coefficients lagging models of heat conduction have already been proposed. In this work, an explicit finite difference scheme for a model with coefficients variable in time is developed, and their properties of convergence and stability are studied. Numerical computations showing examples of applications of the scheme are presented.

Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37

Well-Posedness of Diffusion-Wave Problem with Arbitrary Finite Number of Time Fractional Derivatives in Sobolev Spaces H^s

Mathematics Subject Classification 2010: 26A33, 33E12, 35S10, 45K05.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSFUM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.