Two-dimensional Diffusion Equation Research Articles

Numerical analysis of the Gibbs–Thomson effect on trench-filling epitaxial growth of 4H-SiC

A steady-state two-dimensional diffusion equation was numerically analyzed to examine the rate of homoepitaxial growth on a trenched 4H-SiC substrate. The radii of curvature at the top and bottom of the trenches were used to take the Gibbs–Thomson effect into account in the analysis based on the conventional boundary-layer model. When the trench pitch was less than or equal to 6.0 µm, the measured dependence of the growth rate on the trench pitch was found to be explained by the Gibbs–Thomson effect on the vapor-phase diffusion of growing species.

A three-dimensional refined modeling for the effects of SF6 release in ionosphere

The traditional simulation model of sulfur hexafluoride ionosphere release is a simple point-source model and the simulation precision is not high. The three-dimensional refined simulation model of rocket SF6 release is established in this paper, in which the rocket pose and velocity, gas injection velocity and flow, and wind velocity are all taken into account in the diffusion equation. Meanwhile, the influences of geomagnetic inclination and the field diffusion on artificial disturbance form are considered in the plasma diffusion equation, and the two-dimensional plasma diffusion equation is extended to three-dimensional case. The ray tracing method is used to study the influence of ionospheric artificial disturbance on the short wave propagation path. The research results of the ionosphere kinetics process, ionospheric uneven body generation mechanism and evolution are of great significance.

A Rotated Crank-Nicolson Iterative Method For The Solution of Two-Dimensional Time-Fractional Diffusion Equation

This paper aims to examine the effectiveness of a rotated iterative method to solve the two-dimensional time fractional diffusion equations, which are used when describing transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model. This type of equation is obtained by replacing the first order time derivative in the standard di ff usion equation with a fractional derivative of order α, where 0 < α < 1, in accordance with Riemann-Liouville or Caputo. The developed method is derived from the standard Crank-Nicolson (S(C-N)) scheme by rotating clockwise 45 o with respect to the standard mesh. The study demonstrates the enhanced efficiency and superiority of the rotated Crank-Nicolson (R(C-N)) method, which overall reduces the CPU consumption time.

Three higher order analytical nodal methods for multigroup neutron diffusion equations

This work presents three efficient higher order analytical nodal methods for the numerical solution of a two-dimensional multigroup neutron diffusion equation in Cartesian geometry based on the use of the successive polynomial-weighted transverse integrations technique to convert a one-group diffusion equation to a system of coupled one-dimensional ordinary differential equations. These equations are then solved analytically over each homogenized cell after adequate approximations of the resulting effective sources after transversal integrations. Coupling between the approximate transverse flux-moments is achieved by imposing uniqueness constraint on their moments values. Adjacent elements are coupled by enforcing continuity conditions on the flux and current moments at interfaces cells. The weighted cell-balance equations and current-continuity conditions are then used to derive the discrete equations. These methods are applied for solving numerically various 2D benchmark problems and theirs performances discussed. Numerical results demonstrates more efficiency for the third higher order analytical nodal method for which the alone unknowns considered are the transverse flux moments on the interfaces of the homogenized elements.

Fast and stable explicit operator splitting methods for phase-field models

Numerical simulations of phase-field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods. In this paper, we propose fast and stable explicit operator splitting methods for both one- and two-dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and the Cahn–Hilliard equation. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines together with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. We demonstrate the performance of the proposed methods on a number of one- and two-dimensional numerical examples, where different stages of coarsening such as the initial preparation, alternating rapid structural transition and slow motion can be clearly observed.

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of random walks from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem in numerical experiments to test the models for symmetric operators with spectral qualities similar to light water reactor problems. In general, the derived approximations show good agreement with random walk lengths and leakage fractions computed by the numerical experiments.

Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

AbstractIn this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

An upwind finite volume method on non-orthogonal quadrilateral meshes for the convection diffusion equation in porous media

The aim of this work is to solve the two-dimensional convection diffusion equation on non-rectangular grids formed only by quadrilaterals honoring the internal structures of a reservoir (preferential flow channels, faults, areas of high permeability contrast, changes in sediment type, etc.), taking into account different physical configurations of the porous medium. To take advantage of the good representation of the domain through these meshes, the finite volume method was used, which is conservative and facilitates the treatment of the boundary conditions. In this method, the convection diffusion equation is integrated on each quadrilateral (control volume) of the mesh, thus obtaining the integral form of the equation. The velocity value in the face of each quadrilateral is determined according to the direction of the flow (upwind scheme). After approximating the integrals involved and taking into account the boundary conditions, a discrete equation in each control volume showed up. Finally, a large sparse linear system is obtained, generally non-symmetric and ill-conditioned, which can be solved by iterative methods such as GMRES with incomplete LU preconditioning. Different scenarios were considered varying boundary conditions (Dirichlet and Neumann type), source term, and diffusion constant fluid velocity. The results are consistent with the physical interpretation of each configuration.

Spectral direction splitting methods for two-dimensional space fractional diffusion equations

A numerical method for a kind of time-dependent two-dimensional two-sided space fractional diffusion equations is developed in this paper. The proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization. The direction splitting approach renders the underlying two-dimensional equation into a set of one-dimensional space fractional diffusion equations at each time step. Then these one-dimensional equations are solved by using the spectral method based on weak formulations. A time error estimate is derived for the semi-discrete solution, and the unconditional stability of the fully discretized scheme is proved. Some numerical examples are presented to validate the proposed method.

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Two alternating direction implicit difference schemes are derived for two-dimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete $$L^1(L^\infty )$$L1(L?) norm with the convergence orders $$O(\tau ^2|\ln \tau |+h_1^2+h_2^2+\Delta \alpha ^2)$$O(?2|ln?|+h12+h22+Δ?2) and $$O(\tau ^2|\ln \tau |+h_1^4+h_2^4+\Delta \alpha ^4),$$O(?2|ln?|+h14+h24+Δ?4), respectively, where $$ \tau , h_i \;(i=1,2)$$?,hi(i=1,2) and $$\Delta \alpha $$Δ? are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one- and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative by an integral operator. Some numerical examples show that the convergence orders of the proposed local Pk-DG methods are O(hk+1) both in one and two dimensions, where Pk denotes the space of the real-valued polynomials with degree at most k.

FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part I: Diffusion operator

This is the first in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source MATLAB/GNU Octave toolbox. The intention of this ongoing project is to provide a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix/vector operations and is carefully documented; in addition, a direct mapping between discretization terms and code routines is maintained throughout. The present work focuses on a two-dimensional time-dependent diffusion equation with space/time-varying coefficients. The spatial discretization is based on the local discontinuous Galerkin formulation. Approximations of orders zero through four based on orthogonal polynomials have been implemented; more spaces of arbitrary type and order can be easily accommodated by the code structure.

Methods for reconstruction of the density distribution of nuclear power

In analytical reconstruction method (ARM), the two-dimensional (2D) neutron diffusion equation is analytically solved for two energy groups (2G) and homogeneous nodes with dimensions of a fuel assembly (FA). The solution employs a 2D fourth-order expansion for the axial leakage term. The Nodal Expansion Method (NEM) provides the solution average values as the four average partial currents on the surfaces of the node, the average flux in the node and the multiplying factor of the problem. The expansion coefficients for the axial leakage are determined directly from NEM method or can be determined in the reconstruction method. A new polynomial reconstruction method (PRM) is implemented based on the 2D expansion for the axial leakage term. The ARM method use the four average currents on the surfaces of the node and four average fluxes in corners of the node as boundary conditions and the average flux in the node as a consistency condition. To determine the average fluxes in corners of the node an analytical solution is employed. This analytical solution uses the average fluxes on the surfaces of the node as boundary conditions and discontinuities in corners are incorporated. The polynomial and analytical solutions to the PRM and ARM methods, respectively, represent the homogeneous flux distributions. The detailed distributions inside a FA are estimated by product of homogeneous distribution by local heterogeneous form function. Moreover, the form functions of power are used. The results show that the methods have good accuracy when compared with reference values and the ARM method is more accurate than the PRM method. The largest errors are in the peripheries of FAs, near the reflector, reinforcing the fact that these errors are associated with the homogenization process.

A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

In this article, a new spectral meshless radial point interpolation (SMRPI) technique is proposed and, as a test problem, applied to the two-dimensional diffusion equation with an integral (non-classical) condition. This innovative method is based on erudite combination of meshless methods and spectral collocation techniques but it is not traditional meshless collocation method. As the meshless method, the point interpolation method with the help of radial basis functions is used to construct shape functions as basis functions in the frame of spectral collocation methods. These basis functions (shape functions) have Kronecker delta function property. In the proposed method, high order derivatives (in high order partial differential equations) are evaluated by constructing and using operational matrices. In SMRPI method, it does not require any kind of integration locally over small quadrature domains as it is essential in Galerkin weak form meshless methods such as element free Galerkin (EFG) and meshless local Petrov–Galerkin (MLPG) methods. Also, it is not needed to determine shape parameter which is often played important role in these methods, for instance recall test (weight functions) for moving least squares (MLS) approximation in the frame of MLPG. Therefore, computational costs of SMRPI method is less expensive. A comparison study of the efficiency and accuracy of the present method and meshless local Petrov–Galerkin (MLPG) method is given by applying on mentioned diffusion equation. Convergence studies in the numerical examples show that SMRPI method possesses reasonable rate of convergence.

Pollutant’s Horizontal Dispersion Along and Against Sinusoidally Varying Velocity from a Pulse Type Point Source

An analytical solution of a two-dimensional advection diffusion equation with time dependent coefficients is obtained by using Laplace Integral Transformation Technique. The horizontal medium of solute transport is considered of semi-infinite extent along both the longitudinal and lateral directions. The input concentration is assumed at an intermediate position of the domain. It helps to evaluate concentration level along the flow as well as against the flow through one model only. The source of the input concentration is considered to be of pulse type. In the presence of the source, it is assumed to be decreasing very slowly with time, and just after the elimination of the source it is assumed to be zero. The dispersion coefficient and the advection parameter are considered directly proportional to each other. The analytical solution may be used to predict the solute concentration level with position and time in an open medium as well as in a porous medium. The effect of heterogeneity on the solute transport may also be predicted.

Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.

Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations

Based on the recently developed local radial basis function method, we devise an implicit local radial basis function scheme, which is intrinsic mesh-free, for solving time fractional diffusion equations. In this paper the L1 scheme and the local radial basis function method are applied for temporal and spatial discretization, respectively, in which the time-marching iteration is performed implicitly. The robustness and accuracy of this proposed implicit local radial basis function method are demonstrated by the numerical example. Furthermore, the sensitivities of the shape parameter c and the number of nodes in the local sub-domain to the accuracy of numerical solutions are also investigated.

Physics of Supersonic Mixing in Parallel and Non-Parallel Streams Passing Over Base Thickness

The mixing flow field of two-parallel and non parallel gaseous streams has been studied numerically. The streams are of air and hydrogen and come into contact after passing over a finite thickness base. The two gas streams are delivered from a high-pressure reservoir and entering into the domain with atmospheric pressure. The two-dimensional unsteady Navier-Stokes equations, energy, mass diffusion and species continuity equations are numerically simulated to analyze the mixing layer in supersonic flow field. An explicit Harten-Yee Non-MUSCL Modified flux-type TVD (total variation diminishing) scheme is used to solve the system of equations. An algebraic turbulence model is used to calculate the eddy viscosity coefficient. Keeping constant the inlet pressure and velocity of the streams, the merging angle is varied to observe the physics of flow fields, mixing of two-streams and mixing efficiency. The results show that the increase of merging angle causes stronger interactions between two streams, high momentum exchange and eventually enhancement of mixing between two streams.

A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer

In this work, the local fractional variational iteration method is employed to obtain approximate analytical solution of the two-dimensional diffusion equation in fractal heat transfer with help of local fractional derivative and integral operators.

Laplace variational iteration method for the two-dimensional diffusion equation in homogeneous materials

In this paper, we suggest the local fractional Laplace variational iteration method to deal with the two-dimensional diffusion in homogeneous materials. The operator is considered in local fractional sense. The obtained solution shows the non-differentiable behavior of homogeneous materials with fractal characteristics.