Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Comparison of the interface reaction behaviors of CaO–V_(2)O_(5) and MnO_(2)–V_(2)O_(5) solid-state systems based on the diffusion couple method

The formation mechanism of calcium vanadate and manganese vanadate and the difference between calcium and manganese in the reaction with vanadium are basic issues in the calcification roasting and manganese roasting process with vanadium slag.In this work,CaO–V_(2)O_(5) and MnO_(2)–V_(2)O_(5) diffusion couples were prepared and roasted for different time periods to illustrate and compare the diffusion reaction mechanisms.Then,the changes in the diffusion product and diffusion coefficient were investigated and calculated based on scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDS) analysis.Results show that with the extension of the roasting time,the diffusion reaction gradually proceeds among the CaO–V_(2)O_(5) and MnO_(2)–V_(2)O_(5) diffusion couples.The regional boundaries of calcium and vanadium are easily identifiable for the CaO–V_(2)O_(5) diffusion couple.Meanwhile,for the MnO_(2)–V_(2)O_(5) diffusion couple,MnO_(2) gradually decomposes to form Mn_(2)O_(3),and vanadium diffuses into the interior of Mn_(2)O_(3).Only a part of vanadium combines with manganese to form the diffusion production layer.CaV_(2)O_(6) and MnV_(2)O_(6) are the interfacial reaction products of the CaO–V_(2)O_(5) and MnO_(2)–V_(2)O_(5) diffusion couples,respectively,whose thicknesses are 39.85 and 32.13μm when roasted for 16 h.After 16 h,both diffusion couples reach the reaction equilibrium due to the limitation of diffusion.The diffusion coefficient of the CaO–V_(2)O_(5) diffusion couple is higher than that of the MnO_(2)–V_(2)O_(5) diffusion couple for the same roasting time,and the diffusion reaction between vanadium and calcium is easier than that between vanadium and manganese.

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

基于变分节块法六角形Quasi-diffusion程序开发及验证

传统扩散理论在中子各向异性强的堆芯计算中具有较低的精度,Quasi-diffusion方程相比于传统扩散方程引入更少的近似,通过艾丁顿因子描述传统扩散理论不能反映的中子流各向异性特点。国内外对六角形几何三维Quasi-diffusion方程研究有所不足。针对艾丁顿因子计算的非线性特点,本文基于“两步法”的思想,将艾丁顿因子看作是一个特殊的少群参数,采用中子能谱适应性更好的蒙特卡罗程序SERPENT计算,并对传统扩散变分节块法进行拓展,开发了六角形组件堆芯计算程序VNMQD。采用3D VVER1000基准题、RBWR单组件问题、3D BN600简化模型对程序进行了验证,结果表明:VNMQD程序开发正确,对于非均匀性较强的堆芯,VNMQD比传统扩散方法计算精度更高、计算效率接近,实现了计算精度和计算效率的平衡。

THE FINITE DIFFERENCE STREAMLINE DIFFUSION METHODS FOR TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS

In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L^2-norm.

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

A rapid and accurate direct measurement method of underground coal seam gas content based on dynamic diffusion theory

Coal seam gas content is frequently measured in quantity during underground coal mining operation and coalbed methane(CBM)exploration as a significant basic parameter.Due to the calculation error of lost gas and residual gas in the direct method,the efficiency and accuracy of the current methods are not inadequate to the large area multi-point measurement of coal seam gas content.This paper firstly deduces a simplified theoretical dynamic model for calculating lost gas based on gas dynamic diffusion theory.Secondly,the effects of various factors on gas dynamic diffusion from coal particle are experimentally studied.And sampling procedure of representative coal particle is improved.Thirdly,a new estimation method of residual gas content based on excess adsorption and competitive adsorption theory is proposed.The results showed that the maximum error of calculating the losing gas content by using the new simplified model is only 4%.Considering the influence of particle size on gas diffusion law,the particle size of the collected coal sample is below 0.25 mm,which improves the measurement speed and reflects the safety representativeness of the sample.The determination time of gas content reduced from 36 to 3 h/piece.Moreover,the absolute error is 0.15–0.50 m^3/t,and the relative error is within 5%.A new engineering method for determining the coal seam gas content is developed according to the above research.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.

Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods

In this paper, the cubic and quintic diffusion equation under stochastic non homogeneity is solved using Wiener- Hermite expansion and perturbation (WHEP) technique, Homotopy perturbation method (HPM) and Pickard approximation technique. The analytic solution of the linear case is obtained using Eigenfunction expansion .The Picard approximation method is used to introduce the first and second order approximate solution for the non linear case. The WHEP technique is also used to obtain approximate solution under different orders and different corrections. The Homotopy perturbation method (HPM) is also used to obtain some approximation orders for mean and variance. Using mathematica-5, the methods of solution are illustrated through figures, comparisons among different methods and some parametric studies.

Modeling of Diffusion Transport through Oral Biofilms with the Inverse Problem Method

Aim The purpose of this study was to develop a mathe-matical model to quantitatively describe the passive trans-port of macromolecules within dental biofilms. Methodology Fluorescently labeled dextrans with different molecular mass （3 kD,10 kD,40 kD,70 kD,2 000 kD） were used as a series of diffusion probes. Streptococcus mutans,Streptococcus sanguinis,Actinomyces naeslundii and Fusobacterium nucleatum were used as inocula for biofilm formation. The diffusion processes of different probes through the in vitro biofilm were recorded with a confocal laser microscope. Results Mathematical function of biofilm penetration was constructed on the basis of the inverse problem method. Based on this function,not only the relationship between average concentration of steady-state and molecule weights can be analyzed,but also that between penetrative time and molecule weights. Conclusion This can be used to predict the effective concentration and the penetrative time of anti-biofilm medicines that can diffuse through oral biofilm. Further-more,an improved model for large molecule is proposed by considering the exchange time at the upper boundary of the dental biofilm.

ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS

We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

A new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter e. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally, a numerical experiment is carried out to confirm the theoretical results.

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with the interface correction(DDGIC)(Liu and Yan in Commun Comput Phys 8(3):541-564,2010),the symmetric DDG method(Vidden and Yan in Comput Math 31(6):638-662,2013),and the nonsymmetric DDG method(Yan in J Sci Comput 54(2):663-683,2013).We also include the study of the interior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quantitative errors at the Lobatto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart （CR） element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

Comparison of Numerical Approximations of One-Dimensional Space Fractional Diffusion Equation Using Different Types of Collocation Points in Spectral Method Based on Lagrange’s Basis Polynomials

Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.

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.