Diffusion Problems Research Articles

Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes

Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with p -Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.

Random walk on spheres method for solving anisotropic transient diffusion problems and flux calculations

Abstract The Random Walk on Spheres (RWS) algorithm for solving anisotropic transient diffusion problems based on a stochastic learning procedure for calculation of the exit position of the anisotropic diffusion process on a sphere is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have numerically calculated the probability density for the exit position on a sphere. The first passage time is then represented explicitly. The method can easily be implemented to solve diffusion problems with spatially varying diffusion coefficients for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We apply the developed algorithm for solving an exciton transport in a semiconductor material with a threading dislocation where the measured functions are the exciton fluxes to the semiconductor’s substrate and on the dislocation surface.

Fast-reaction limit of reaction–diffusion systems with nonlinear diffusion

In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form [Formula: see text] in usual reaction–diffusion equation, which represent linear diffusion, by terms of form [Formula: see text], representing nonlinear diffusion. We prove the convergence in the fast-reaction limit [Formula: see text] that is determined by the unique solution of a certain scalar nonlinear diffusion problem.

Hybrid method for singularly perturbed Robin type parabolic convection–diffusion problems on Shishkin mesh

This work presents a numerical solution to singularly perturbed Robin-type parabolic convection–diffusion problems. A hybrid method that combines the central difference scheme in the inner region and the midpoint of the upwind scheme in the outer region on a Shishkin mesh is utilized in the space direction, and an implicit Euler technique is used in the time direction. The analysis of convergence demonstrates that the present method converges with O(N−2ln2N+Δt). The theoretical findings are supported by two numerical examples. The upwind method proposed in the literature is improved by the present method.

Fast successive permutation iterative algorithms for solving convection diffusion equation

The solution of convection diffusion problem is a complex task, requiring the solving of linear equations which can take up a significant amount of computational time. This issue can be mitigated through the use of iterative methods. We present a class of fast successive permutation iterative algorithms with relaxation factors (ω-SPI) to solve 1D, 2D and 3D convection diffusion equations. The stability, convergence and optimal value of the relaxation factor are proved. In order to test the efficiency and accuracy of our proposed iterative algorithms, we present several numerical experiments. The results demonstrate that the new algorithms are more accurate and faster than traditional methods such as full-implicit, Gauss-Seidel and SOR algorithms.

Performances analysis of parallel asynchronous and synchronous algorithms on cloud architecture for PDE

In order to solve discretized stationary linear or non linear diffusion and convection–diffusion problems, the present paper deals with a comparison between the performances of parallel iterative synchronous or asynchronous algorithms implemented on a Cloud computing architecture with several sizes of the algebraic systems to be solved. The parallel algorithms are briefly and intuitively presented and analysis of their behavior is recalled. In particular, criteria of convergence are recalled and estimates of convergence speed are given. Implementation on cloud architectures are briefly described. Finally, for each kind of problem, the performance of the target algorithms on cloud architectures are analyzed.

An adaptive global–local generalized FEM for multiscale advection–diffusion problems

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEMgl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEMgl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEMgl closely matches reference fine-scale FEM solutions in the L∞ norm.

Interplay between Brownian and hydrodynamic tracer diffusion in suspensions of swimming microorganisms

The general problem of tracer diffusion in non-equilibrium baths is important in a wide range of systems, from the cellular level to geographical length scales. In this paper, we revisit the archetypical example of such a system: a collection of small passive particles immersed in a dilute suspension of non-interacting dipolar microswimmers, representing bacteria or algae. In particular, we consider the interplay between thermal (Brownian) diffusion and hydrodynamic (active) diffusion due to the persistent advection of tracers by microswimmer flow fields. Previously, it has been argued that even a moderate amount of Brownian diffusion is sufficient to significantly reduce the persistence time of tracer advection, leading to a significantly reduced value of the effective active diffusion coefficient $D_A$ compared to the non-Brownian case. Here, we show by large-scale simulations and kinetic theory that this effect is in fact practically relevant only for microswimmers that effectively remain stationary while still stirring up the surrounding fluid – so-called shakers. In contrast, for moderate and high values of the swimming speed, relevant for biological microswimmer suspensions, the effect of Brownian motion on $D_A$ is negligible, leading to the effects of advection by microswimmers and Brownian motion being additive. This conclusion contrasts with previous results from the literature, and encourages a reinterpretation of recent experimental measurements of $D_A$ for tracer particles of varying size in bacterial suspensions.

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

The Progress of the Interfacial Diffusion between Virgin and Aged Asphalt Based on Molecular Dynamics Simulation: A Review

The utilization rate of reclaimed asphalt pavement is significantly low in the sustainable design process of asphalt roads. Numerous researchers have extensively investigated the recycling and utilization of reclaimed asphalt pavements from various perspectives. Molecular dynamics simulations could elucidate the diffusion phenomenon occurring at the molecular scale between virgin and aged asphalt interfaces. This review provides a comprehensive summary of the simulation methods and applications of molecular dynamics simulation in the interface diffusion problem between virgin and aged asphalt. Diffusion theory and model testing methods are discussed. The review proposes the basic steps of molecular dynamics simulation and summarizes the molecular models with the corresponding parameter settings of virgin asphalt and aged asphalt. Moreover, the current influencing factors on the interfacial diffusion problem of virgin and aged asphalt are discussed. The paper explores the validation parameters including density, viscosity, radial distribution function (RDF), glass transition temperature, and solubility parameters based on the existing research. Molecular dynamics simulation could simulate interface diffusion at a micro-scale and clarify the diffusion depth and influence range of different asphalts. The purpose of the study of molecular dynamics is to solve interface issues and advance optimization of reclaimed materials.

High-order spectral collocation method using tempered fractional Sturm–Liouville eigenproblems

This paper presents an accurate exponential tempered fractional spectral collocation method (TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly, we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional advection and diffusion problem, the time-space tempered fractional advection–diffusion problem (TFADP), the multi-term time-space tempered fractional problems, and the time-space tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability of the fractional collocation method. The study includes a numerical examination of the produced condition number κ(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa (A)$$\\end{document} of the linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-norm error and exponential rate of spectral convergence.

Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution using higher-order continuous B-spline basis functions in both spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.

Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh in 2D

As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolant is introduced. In fact, this interpolant is composed of a vertices-edges-element interpolant within the layer and a local L2-projection outside the layer. On the basis of that, by selecting penalty parameters on different types of interelement edges, we further obtain supercloseness of almost k+12 order in an energy norm. Here k is the degree of piecewise polynomials. Numerical tests support our theoretical conclusion.

High order positivity-preserving nodal discontinuous Galerkin methods for anisotropic diffusion problems

In this paper, we develop second and third order accurate positivity-preserving (PP) nodal discontinuous Galerkin (DG) methods for one and two dimensional anisotropic diffusion problems. The key idea is to first represent the cell average of its numerical approximation as a weighted summation of Gaussian quadrature point values used in the updating of nodal DG methods, and then transform these Gaussian quadrature point values to some other special chosen point values. We prove that by taking parameters in the definition of nodal DG methods appropriately, together with a suitable time stability condition, the cell averages can be kept positive. A polynomial scaling limiter is then applied to obtain positive numerical approximations on the whole cell without sacrificing accuracy. Stability analysis without the PP limiter is also rigorously established. Numerical experiments are performed to demonstrate desired orders of accuracy, PP and good performances of our proposed approach.

Dynamic behavior of reaction–diffusion equation with Holling II predation

In this paper, we consider the positive solutions for some reaction–diffusion equations with Holling II predation. The main aim is to analyze the long time behavior of solutions, both the diffusion and non-diffusion problems are investigated. Our results also exhibit different effects between Dirichlet and Neumann boundary conditions.

Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. Mathematics subject classification03B80, 65L10, 65L12

I solution of the heat conduction problem for computational modeling of the hydrogen content reduction in large-size forgings during antiflocken heat processing

The paper investigates problems of flocken formation in large-sized billets during heat treatment increasing hydrogen degassing. The solution of thermal conductivity problem is given, the estimation of thermal cycles at preliminary heat treatment for the subsequent solution of hydrogen diffusion problem is carried out on the example of medium-alloyed steels of 5KhNM and 20Kh3NMFA grades.

Numerical treatment of reactive diffusion using the discontinuous Galerkin method

AbstractThis work presents a new finite element variational formulation for the numerical treatment of diffusional phase transformations using the discontinuous Galerkin method (DGM). Steep concentration and property gradients near phase boundaries require particular focus on a sound numerical treatment. There are different ways to tackle this problem ranging from (i) the well-known phase field method (PFM) (Biner et al. in Programming phase-field modeling, Springer, Berlin, 2017, Emmerich in The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, Springer, Berlin, 2003), where the interface is described continuously to (ii) methods that allow sharp transitions at phase boundaries, such as reactive diffusion models (Svoboda and Fischer in Comput Mater Sci 127:136–140, 2017, 78:39–46, 2013, Svoboda et al. in Comput Mater Sci 95:309–315, 2014). Phase transformation problems with continuous property changes can be implemented using the continuous Galerkin method (GM). Sharp interface models, however, lead to stability problems with the GM. A method that is able to treat the features of sharp interface models is the discontinuous Galerkin method. This method is well understood for regular diffusion problems (Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003). As will be shown, it is also particularly well suited to model phase transformations. We discuss the thermodynamic background by review of a multi-phase, binary system. A new DGM formulation for the phase transformation problem with sharp interfaces is then introduced. Finally, the derived method is used in a 2D microstructural evolution simulation that features a binary, three-phase system that also takes the vacancy mechanism of solid body diffusion into account.