Diffusion Equation Method Research Articles

Speckle Noise Suppression Based on Empirical Mode Decomposition and Improved Anisotropic Diffusion Equation

Existing methods to eliminate the laser speckle noise in quantitative phase imaging always suffer from the loss of detailed phase information and the resolution reduction in the reproduced image. To overcome these problems, this paper proposes a speckle noise suppression method based on empirical mode decomposition. Our proposed method requires only one image without additional equipment and avoids the complicated process of searching the optimal processing parameters. In this method, we use empirical mode decomposition to highlight the high frequency information of the interference image and use the Canny operator to perform edge detection, so the diffusion denoising process is guided by high-precision detection results to achieve better results. To validate the performance of our proposed method, the phase maps processed by our proposed method are compared with the phase maps processed by the improved anisotropic diffusion equation method with edge detection, the mean filter method and the median filter method. The experimental results show that the method proposed in this paper not only has a better denoising effect but also preserves more details and achieves higher phase reconstruction accuracy.

A Discontinuous Galerkin Finite Element Method based axial SN for the 2D/1D transport method

The 2D/1D transport method is a prominent solver for the 3D Boltzmann equation due to its strong geometric capability. To pursue high accuracy, the discrete ordinate (SN) method for transport equation is a straightforward choice as the axial 1D solver instead of the traditional nodal expansion method (NEM) for diffusion equation. In practice, the axial SN solver is usually combined with the transverse leakage splitting method to deal with the negative source issue, which strongly affects the convergence of the 2D/1D method. However, transverse leakage splitting breaks the consistency of scalar fluxes between the 2D MOC and 1D SN equations. To solve this problem, a discontinuous Galerkin finite element method (DGFEM) for axial SN together with an improved 2D/1D transport approach is proposed in the present study to ensure that the scalar flux from the 1D axial solver is the same as that from the 2D solver. The accuracy, consistency and computational efficiency of the new scheme are evaluated using various well-known numerical problems, including heterogeneous assembly problem, KUCA benchmark, and C5G5-3D benchmark.

A New Modified Technique of Adomian Decomposition Method for Fractional Diffusion Equations with Initial-Boundary Conditions

In general, solving fractional partial differential equations either numerically or analytically is a difficult task. However, mathematicians have tried their best to make the task easy and promoted various techniques for their solutions. In this regard, a very prominent and accurate technique, which is known as the new technique of the Adomian decomposition method, is developed and presented for the solution of the initial-boundary value problem of the diffusion equation with fractional view analysis. The suggested model is an important mathematical model to study the behavior of degrees of memory in diffusing materials. Some important results for the given model at different fractional orders of the derivatives are achieved. Graphs show the obtained results to confirm the accuracy and validity of the suggested technique. These results are in good contact with the physical dynamics of the targeted problems. The obtained results for both fractional and integer orders problems are explained through graphs and tables. Tables and graphs support the physical behavior of each problem and the best of physical analysis. From the results, it is concluded that as the fractional order derivative is changed, the graphs or paths of dynamics are also changed. Therefore, we now choose the best solution or dynamic of the problem at a particular derivative order. It is analyzed that the present technique is one of the best techniques to handle the solutions of fractional partial differential equations having initial and boundary conditions (BCs), which are very rare in literature. Furthermore, a small number of calculations are done to achieve a very high rate of convergence, which is the novelty of the present research work. The proposed method provides the series solution with twice recursive formulae to increase the desired accuracy and is preferred among the best techniques to find the solution of fractional partial differential equations with mixed initials and BCs.

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation

This paper presents an effective numerical method for multi-term variable-order time fractional diffusion equations with the variable-order fractional derivative. The local discontinuous Galerkin method and the finite difference method are used in the spatial and temporal directions, respectively. We prove that the scheme is unconditional stable and convergent with O(hs+1+(Δt)2−r), where r=max{ɛ(t)}. s, h, Δt are the degree of piecewise polynomials, the space step sizes, and the time step sizes, respectively. Some numerical experiments are used to illustrate the effectiveness and applicability of the scheme.

A numerical method for a nonlocal diffusion equation with additive noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

Nonequilibrium quantum thermodynamics in non-Markovian adiabatic speedup

Understanding heat transfer between a quantum system and its environment is of undisputed importance if reliable quantum devices are to be constructed. Here, we investigate the heat transfer between system and bath in non-Markovian open systems in the process of adiabatic speedup. Using the quantum state diffusion equation method, the heat current, energy current, and power are calculated during free evolution and under external control of the system. While the heat current increases with increasing system–bath coupling strength and bath temperature, it can be restricted by the non-Markovian nature of the bath. Without pulse control, the heat current is nearly equal to the energy current. On the other hand, with pulse control, the energy current turns out to be nearly equal to the power. In this scenario, we show that non-Markovianity is a useful tool to drive the system through an approximate adiabatic dynamics, with pulse control acting in the conversion between heat current and power throughout the evolution.

Connection between the existence of a priori estimate for a flux and the convergence of iterative methods for diffusion equation with highly varying coefficients

Abstract An iterative method with the number of iterations independent of the coefficient jumps is proposed for the boundary value problem for a diffusion equation with highly varying coefficient. The method applies one solution of the Poisson equation at each step of iteration. In the present paper we extend the class of domains the iterative method is justified for.

A Class of Unconditioned Stable 4-Point WSGD Schemes and Fast Iteration Methods for Space Fractional Diffusion Equations

A Class of Unconditioned Stable 4-Point WSGD Schemes and Fast Iteration Methods for Space Fractional Diffusion Equations

An Efficient MultiModes Monte Carlo Homogenization Method for Random Materials

In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample $\omega$, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting in a random diffusion equation. In the second stage, the resulting random diffusion equation is approximated and computed by using an efficient multimodes Monte Carlo method which only requires solving a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established, and the optimal rate of convergence is also obtained for the proposed multimodes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.

Partially explicit time discretization for nonlinear time fractional diffusion equations

Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses a coarse spatial grid for spatial resolution. For temporal discretization, implicit methods are often used. For implicit methods, though the time step can be relatively large, the equations are difficult to compute due to the nonlinearity and the fact that one deals with large-scale systems. On the other hand, the discrete systems in explicit methods are easier to compute but it requires small time steps. In this work, we propose a partially explicit scheme following earlier works on developing partially explicit methods for nonlinear diffusion equations. In these schemes, the diffusion term is treated partially explicitly and the reaction term is treated fully explicitly. With the appropriate construction of spaces and stability analysis, we find that the required time step in our proposed scheme scales as the coarse mesh size, which creates a computational saving. The main novelty of this work is the extension of our earlier works for diffusion equations to time fractional diffusion equations. For the case of fractional diffusion equations, the constraints on time steps are more severe and the proposed methods alleviate this problem since the time step in the partially explicit method scales as the coarse mesh size. We present stability analysis. Numerical results are presented where we compare our proposed partially explicit methods with a fully implicit approach. We show that our proposed approach provides similar results while treating many degrees of freedom in nonlinear terms explicitly.

On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper Dimension-adaptive tensor-product quadratuture Computing (2003) by T. Gerstner and M. Griebel) is explored.

General linear and spectral Galerkin methods for the nonlinear two-sided space distributed-order diffusion equation

A high order numerical method is constructed for the two-dimensional nonlinear two-sided space distributed-order diffusion equation, in which the distributed-order integral is approximated by the Gauss-Legendre quadrature formula, the time and space are discretized by the general linear method and spectral Galerkin method, respectively. The proposed method is proved to be stable and convergent of order p in time, when the nonlinear term satisfies the local Lipschitz condition and the general linear method with generalized stage order p is coercive and algebraically stable. Moreover, the optimal error estimates in distributed-order integral and in space are also obtained. Finally, numerical experiments are presented to illustrate the theoretical estimates.

A lopsided scaled DTS preconditioning method for the discrete space-fractional diffusion equations

The Grünwald–Letnikov type formulas are applied to discretize the space-fractional diffusion equation and the discretization result is got as a system of linear equations. A lopsided scaled diagonal and Toeplitz splitting (LSDTS) iteration method is constructed to solve this linear system and the convergence is also discussed. An LSDTSτ preconditioner is proposed based on the LSDTS iteration method and the GMRES method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the LSDTSτ preconditioned matrix are clustered. The numerical experiments illustrate that the LSDTSτ preconditioner can significantly improve the convergence properties of the GMRES method.

A fast finite volume method for spatial fractional diffusion equations on nonuniform meshes

In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel xα−1,α∈(0,1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O(mlog2⁡m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm.

Fast and efficient finite difference method for the distributed-order diffusion equation based on the staggered grids

In this article, an efficient block-centered finite difference scheme is constructed to solve the distributed-order diffusion equation with Neumann boundary condition on the rectangular grids. We use the mid-rectangle formula to approximate the integral term and a fast technique to evaluate the fractional derivative to derive a valid and efficient scheme. The stability and convergence of the efficient scheme for both pressure and velocity are proved rigorously. Finally, we carry out some numerical examples to verify our theoretical analysis and present the efficiency of our scheme.

Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high-resolution compact discretization

The present method describes the high-resolution compact discretization method for the numerical solution of the nonlinear fractal convection-diffusion model on a rectangular plate by employing the Hausdorff distance metric. Estimation of anomalous diffusion is formulated by averaging forward and backward mesh stencils. The higher-order fractional derivatives are appropriately approximated on a minimum mesh stencil and subsequently considered for designing a numerical method that falls in the scope of expanded accuracy. Compact discretization is an efficient technique for partial differential equations; however, studies that apply high-resolution scheme for fractional-order systems are still uninvestigated. A second and fourth-order numerical method for the fractional-order convection-dominated anomalous diffusion equation in two dimensions is constructed for practical applications. Convergence of high-order method is obtained for the nonlinear partial differential equations employing Hausdorff fractal distance metric. The numerical simulations with fractal Graetz-Nusselt equation, fractal Poisson equation, fractal Schrödinger equation, and anomalous diffusion equations with variable and constant coefficients are considered to illustrate the utility of the numerical method in the context of local fractional partial differential equations.•The paper demonstrates a computational method for the fractal convection-diffusion model on a rectangular plate.•Two numerical methods of order two and four for the mildly nonlinear fractional-order convection-dominated anomalous diffusion equations are proposed.•The high-resolution scheme is computationally efficient and makes use of minimal data storage.Method name: High-order method for 2D convection-dominated anomalous diffusion equation, Graetz-Nusselt equation, Poisson equation, and Schrödinger equation in fractal media

Galerkin spectral method for a multi-term time-fractional diffusion equation and an application to inverse source problem

<abstract><p>In this paper, we employ the Galerkin spectral method to handle a multi-term time-fractional diffusion equation, and investigate the numerical stability and convergence of the proposed method. In addition, we find an interesting application of the Galerkin spectral method to solving an inverse source problem efficiently from the noisy final data in a general bounded domain, and the uniqueness and the ill-posedness for the inverse problem are proved based on expression of the solution. Furthermore, we compare the calculation results of spectral method and finite difference method without any regularization method, and get a norm estimate of the coefficient matrix of a spectral method discretized. And for that we conclude that the spectral method itself can act as a regularization method for some inverse problem (such as inverse source problem). Finally, several numerical examples are used to illustrate the effectiveness and accuracy of the method.</p></abstract>

An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

<abstract><p>In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.</p></abstract>

The Corrected Finite Volume Element Methods for Diffusion Equations Satisfying Discrete Extremum Principle

The Corrected Finite Volume Element Methods for Diffusion Equations Satisfying Discrete Extremum Principle