Numerical Discretization Scheme Research Articles

Energy dissipation and evolutions of the nonlocal Cahn-Hilliard model and space fractional variants using efficient variable-step BDF2 method

In this work, an energy stable BDF2 scheme with general nonuniform time steps is developed for the nonlocal Cahn-Hilliard model to capture the multi-scale behavior of evolution efficiently and accurately. The fully discrete numerical scheme is demonstrated to inherit main physical properties of the continuous model, namely, mass conservation and energy dissipation. Recently, various variants of the Cahn-Hilliard model with nonlocal operators and fractional Laplacian have attracted great attention in terms of theoretical analysis and numerical approximation, while there is less emphasis on the comparisons among the different variant models. By comparing the impact of various parameters and coarsening behaviors, numerical results suggest that the nonlocal Cahn-Hilliard model bears a strong resemblance to the space fractional Cahn-Hilliard model derived from H−1 gradient flow of the nonlocal energy. Nevertheless, the order of the fractional Laplacian appears to affect only the rate of energy decay, without altering the profile of the solution for the fractional model relating to the H−β (0<β<1) gradient flow of the classical Ginzburg-Landau energy functional in one dimensional case.

Numerical simulation of the ferrohydrodynamics flow using an unconditionally stable second‐order scheme

AbstractIn this paper, we design a decoupled, linear, unconditionally stable and fully discrete numerical scheme for a ferrohydrodynamics system with second‐order temporal accuracy. This scheme is based on a second‐order backward difference formula for time derivative terms and linearization extrapolation for nonlinear terms, which produces a series of decoupled linear equations and solves effectively this nonlinear and multiphysical coupled system. Meanwhile, we show that the scheme is unconditionally stable. Finally, some numerical experiments are provided to verify the theoretical finding and illustrate the accuracy and efficiency of the proposed scheme.

Learning the flux and diffusion function for degenerate convection-diffusion equations using different types of observations

In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions.

Optimal control problem with nonlinear fractional system constraint applied to image restoration

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on ‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model.

Design and Application of DG-FEM Basis Functions for Neutron Transport on Two-Dimensional and Three-Dimensional Hexagonal Meshes

Reactor design requires safety studies to ensure that the reactors will behave appropriately under incidental or accidental situations. Safety studies often involve multiphysics simulations where several branches of reactor physics are necessary to model a given phenomenon. In those situations, it has been observed that the neutron transport part is still a bottleneck in terms of computational times, with more than 80% of the total time. In the case of hexagonal lattice reactors, transport solvers usually invert the discretised Boltzmann equation by discretising the regular hexagon into lozenges or triangles. In this work, we seek to reduce the computational burden of the neutron transport solver by designing a numerical spatial discretisation scheme that would be more appropriate for honeycomb meshes. In our past research efforts, we have set up interesting discretisation schemes in the finite element setting in 2D, and we wish to extend them to 3D geometries that are prisms with a hexagonal base. In 3D, a rigorous method was derived to shrink the tensor product between 2D and 1D bases to minimum terms. We have applied these functions successfully on a reactor benchmark—Takeda Model 4—to compare and contrast the numerical results in a physical setting.

Variational and Numerical Analysis for an Electro-Viscoelastic Unilateral Contact Problem with Adhesion

In this paper we investigate a mathematical model for the unilateral contact between an electro-viscoelastic body and a conductive foundation. The Signorini conditions are used to model the contact, and adhesion between the contact surfaces is also taken into account. We establish the existence and uniqueness of a weak solution to this problem. Furthermore, we propose a fully discrete numerical scheme to approximate the solution, and we present and prove the main result concerning the estimation of errors.

A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy.

Linear and unconditionally energy stable schemes for the modified phase field crystal equation

In this paper, two temporal discrete numerical schemes of the modified phase field crystal equation are presented by introducing a Lagrange multiplier U and two auxiliary functions ψ, W, which reduce the order of the original equation in time and space. For these schemes, we prove their mass conservation and energy stability. Error estimates of the BDF2 scheme are given. In the end, some numerical experiments are used to verify the correctness of our theoretical results.

Numerical analysis of a second-order energy-stable finite element method for the Swift-Hohenberg equation

In this work, we first prove the existence, uniqueness and regularity of the solution of the Swift-Hohenberg equation by applying the Galerkin spectral method. Then we investigate the convergence of a finite element method in the mixed formulation for the Swift-Hohenberg equation, with Crank-Nicolson scheme in time discretization. We prove that our semidiscrete and fully discrete numerical schemes satisfy unique solvability and unconditional energy stability. Moreover, we prove optimal error estimates for the schemes. Finally, numerical tests are given to validate our theoretical results.

Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme.

A SAV finite element method for the Cahn–Hilliard equation with dynamic boundary conditions

In this paper, we propose a SAV method for the Cahn–Hilliard type phase field model with new dynamic boundary conditions. By using the continuous finite element method in space and the backward difference method in time, the fully discrete numerical schemes preserving the energy law are constructed. Numerical examples show that the proposed scheme can simulate the phase field model well even in a relatively rough grid.

Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciences

We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems (viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.

Second‐order difference scheme for the time fractional mixed diffusion‐wave equation with initial weak regularity

An L1‐type difference scheme on graded meshes for time fractional mixed diffusion‐wave equation with two Caputo derivatives of fractional order and is established and analyzed. In particular, the solution of governing equation typically exhibits a weak regular behavior at an initial time. A fully discrete numerical scheme based on finite difference method is proposed and detailed theoretical analysis for the scheme is also investigated. The novelty of this paper is to show that the L1‐type difference scheme for the model problem under consideration is temporal second‐order accurate over graded meshes. Numerical tests are presented to support our theoretical results.

Feedback control for fluid mixing via advection

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂Rd, with d=2 or d=3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H1(Ω))′ of H1(Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.

Fully discrete, decoupled and energy-stable Fourier-Spectral numerical scheme for the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy flow regime of two-phase incompressible flows

In this paper, we introduce fully discrete Fourier-Spectral numerical scheme to solve the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy equations, which represent a phase-field model for two-phase incompressible flow in either the free flow regime or a Hele-Shaw cell. The proposed scheme achieves full decoupling while maintaining linearity and energy stability through a combination of the Scalar Auxiliary Variable (SAV) method, which discretizes the nonlinear potential, and the “Zero-Energy-Contribution” (ZEC) method, which handles the coupled nonlinear advective/surface tension terms. The efficiency of this scheme is attributed to its linear decoupling structure and the fact that it requires only a few elliptic equations with constant coefficients to be solved at each time step. We rigorously establish the scheme’s unconditional energy stability. Further, some numerical simulations are provided in both 2D and 3D to show its effectiveness, including its accuracy, stability, and efficiency.

Unifying incidence and prevalence under a time-varying general branching process

Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump–Mode–Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman–Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.

FEM–CM as a hybrid approach for multiscale modeling and simulation of ferroelectric boundary value problems

Constitutive modeling of ferroelectrics is a challenging task, spanning physical processes on different scales from unit cell switching and domain wall motion to polycrystalline behavior. The condensed method (CM) is a semi-analytical approach, which has been efficiently applied to various problems in this context, ranging from self-heating and damage evolution to energy harvesting. Engineering applications, however, inevitably require the solution of arbitrary boundary value problems, including the complex multiphysical constitutive behavior, in order to analyze multifunctional devices with integrated ferroelectric components. The well-established finite element method (FEM) is commonly used for this purpose, allowing sufficient flexibility in model design to successfully handle most tasks. A restricting aspect, especially if many calculations are required within, e.g., an optimization process, is the computational cost which can be considerable if two or even more scales are involved. The FEM–CM approach, where a numerical discretization scheme for the macroscale is merged with a semi-analytical methodology targeting at material-related scales, proves to be very efficient in this respect.

Accuracy Estimates for Bilinear Optimal Control Problems Governed by Ordinary Differential Equations

First- and second-order accuracy estimates for an optimal control problem governed by a system of ordinary differential equations with a bilinear control mechanism are presented. The numerical time discretization scheme under consideration is the finite element method with continuous piecewise linear functions. Central to this work is a first- and second-order analysis of optimality of the continuous and approximated optimal control problems. In the case of box constraints on the control, first-order error estimates for the control function are obtained assuming a piecewise constant approximation of the control, whereas second-order accuracy can be obtained in the case of a continuous, piecewise polynomial approximation. Numerical evidence is presented that supports the theoretical findings.

Multiscale Coupled Models for Complex Media: From Analysis to Simulation in Geophysics and Medicine

Many real-life applications require mathematical models at multiple scales, defined in domains with complex structures, some of which having time dependent boundaries. Mathematical models of this type are encountered in seemingly disparate areas e.g., flow and deformation in the subsurface or beneath the ocean floor, and in processes of clinical relevance. While the areas are different, the structure of the models and the challenges are shared: the analysis and simulation must account for the evolution of the domain due to the many coupled processes in the multi-scale context. The key theme and focus of the workshop were novel ideas in the mathematical modeling, analysis, and numerical simulation, which are cross-cutting between the two application areas mentioned above. The talks have covered the mathematical treatment of such problems, as well as the development of efficent numerical discretization schemes and of solvers for large-scale problems.

An Application of the Distributed-Order Time- and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y-direction). In this work, we propose a distributed-order time- and space-fractional diffusion-wave equation to model a comb structure in the more general setting. The distributed-order time- and space-fractional diffusion-wave equation is firstly formulated to study the anomalous diffusion in the comb model subject to an irregular convex domain with the motivation that the time-fractional derivative considers the memory characteristic and the space one with the variable diffusion coefficient possesses the nonlocal characteristic. The finite element method is applied to obtain the numerical solution. The stability and convergence of the numerical discretization scheme are derived and analyzed. Two numerical examples of relevance to the comb model are given to verify the correctness of the numerical method. Moreover, the influence of the involved parameters on the three-dimensional and axial projection drawing particle distribution subject to an elliptical domain are analyzed, and the physical meanings are interpreted in detail.