Semi-discrete Numerical Scheme Research Articles

On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.

Discontinuous Galerkin Methods with Generalized Numerical Fluxes for the Vlasov-Viscous Burgers’ System

In this paper, semi-discrete numerical scheme for the approximation of the periodic Vlasov-viscous Burgers’ system is developed and analyzed. The scheme is based on the coupling of discontinuous Galerkin approximations for the Vlasov equation and local discontinuous Galerkin approximations for the viscous Burgers’ equation. Both these methods use generalized numerical fluxes. The proposed scheme is both mass and momentum conservative. Based on generalized Gauss–Radau projections, optimal rates of convergence in the case of smooth compactly supported initial data are derived. Finally, computational results confirm our theoretical findings.

A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

<abstract><p>We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.</p></abstract>

The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

<abstract><p>In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the $ L^2 $ convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.</p></abstract>

Sparse Monte Carlo Method for Nonlocal Diffusion Problems

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media, peridynamics, and models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. Our method requires minimal assumptions on the regularity of the data. In particular, the kernel of the nonlocal diffusivity is assumed to be a square integrable function and may be singular or discontinuous. An important feature of our method is the use of sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows us to use fewer discretization points without compromising the accuracy. For kernels with singularities, more points are selected automatically in the regions near the singularities. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations, respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

The time-fractional Cattaneo equation is an equation where the fractional order α∈(1,2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme.

A semi-discrete numerical scheme for nonlocally regularized KdV-type equations

A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied by the addition of a linear convolution term involving third-order derivative. To solve the Cauchy problem we propose a semi-discrete numerical method based on a uniform spatial discretization, that is an extension of a previously published work of the present authors. We prove uniform convergence of the numerical method as the mesh size goes to zero. We also prove that the localization error resulting from localization to a finite domain is significantly less than a given threshold if the finite domain is large enough. To illustrate the theoretical results, some numerical experiments are carried out for the Rosenau-KdV equation, the Rosenau-BBM-KdV equation and a convolution-type integro-differential equation. The experiments conducted for three particular choices of the kernel function confirm the error estimates that we provide.

Structure-Preserving Numerical Approximations to a Non-isothermal Hydrodynamic Model of Binary Fluid Flows

We present two second order, structure-preserving numerical schemes for a newly derived thermodynamically consistent, non-isothermal hydrodynamical phase field model for incompressible binary viscous fluid flows. The schemes preserve the volume of each fluid phase, the total energy and the positive entropy production rate. The entropy quadratization approach is employed to devise the two semi-discrete numerical schemes in time, preserving both the total energy and the positive entropy production rate. The first scheme is weakly nonlinear, which is solved using iterative methods aided by fast Fourier algorithms. The second scheme is linear, in which a time-dependent supplementary variable is added to preserve the positive entropy production rate. The semi-discrete schemes are discretized in space by a finite difference method on staggered grids subsequently to yield two fully discrete schemes. Mesh refinement is carried out to confirm the order of the schemes and several numerical examples are provided to show hydrodynamic as well as thermal effects in resolving thermocapillary convection near the fluid interface in the incompressible binary viscous fluid flow.

Energy-stable predictor–corrector schemes for the Cahn–Hilliard equation

In this paper, we construct a new class of predictor–corrector time-stepping schemes for the Cahn–Hilliard equation, which are linear, second-order accurate in time, unconditionally energy stable, and uniquely solvable. Then, we present the stability and error estimates of the semi-discrete numerical schemes for solving the Cahn–Hilliard equation with general nonlinear bulk potentials. The semi-discrete scheme is further discretized using the compact central finite difference method. Several numerical examples are shown to verify the theoretical results. In particular, the numerical simulations show that the predictor–corrector schemes reach the second-order convergence rate at relatively larger time-step sizes than the classical linear schemes. The numerical strategies and theoretical tools developed in this article could be readily applied to study other phase-field models or models that can be cast as gradient flow problems.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations

The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

Fractional Gray–Scott model: Well-posedness, discretization, and simulations

The Gray–Scott (GS) model represents the dynamics and steady state pattern formation in reaction–diffusion systems and has been extensively studied in the past. In this paper, we consider the effects of anomalous diffusion on pattern formation by introducing the fractional Laplacian into the GS model. First, we prove the well-posedness of the fractional GS model. We then introduce the Crank–Nicolson (C–N) scheme for time discretization and weighted shifted Grünwald difference operator for spatial discretization. We perform stability analysis for the time semi-discrete numerical scheme, and furthermore, we analyze numerically the errors with benchmark solutions that show second-order convergence both in time and space. We also employ the spectral collocation method in space and C–N scheme in time to solve the GS model in order to verify the accuracy of our numerical solutions. We observe the formation of different patterns at different values of the fractional order, which are quite different from the patterns of the corresponding integer-order GS model, and quantify them by using the radial distribution function (RDF). Finally, we discover the scaling law for steady patterns of the RDFs in terms of the fractional order.

An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems

As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in Zhang (2017) [42] in many scenarios. Hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

Optimizing the Kelvin force in a moving target subdomain

In order to generate a desired Kelvin (magnetic) force in a target subdomain moving along a prescribed trajectory, we propose a minimization problem with a tracking type cost functional. We use the so-called dipole approximation to realize the magnetic field, where the location and the direction of the magnetic sources are assumed to be fixed. The magnetic field intensity acts as the control and exhibits limiting pointwise constraints. We address two specific problems: the first one corresponds to a fixed final time whereas the second one deals with an unknown force to minimize the final time. We prove existence of solutions and deduce local uniqueness provided that a second-order sufficient condition is valid. We use the classical backward Euler scheme for time discretization. For both problems we prove the [Formula: see text]-weak convergence of this semi-discrete numerical scheme. This result is motivated by [Formula: see text]-convergence and does not require second-order sufficient condition. If the latter holds then we prove [Formula: see text]-strong local convergence. We report computational results to assess the performance of the numerical methods. As an application, we study the control of magnetic nanoparticles as those used in magnetic drug delivery, where the optimized Kelvin force is used to transport the drug to a desired location.

A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.

Semi-Discrete Galerkin Finite Element Method for the Diffusive Peterlin Viscoelastic Model

AbstractIn this paper, a semi-discrete Galerkin finite element method is applied to the two-dimensional diffusive Peterlin viscoelastic model which can describe the unsteady behavior of some incompressible ploymeric fluids. For the derived semi-discrete finite element spatial discretization scheme, the a priori bounds are given that does not rely on the mesh width restriction. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented, respectively. Finally, in order to implement the proposed semi-discrete numerical scheme, we derive three kinds of fully discrete schemes, e.g., Newton’s iterative scheme, Picard’s iterative scheme and implicit-explicit time-stepping scheme. Finally, several numerical experiments are conducted to confirm our theoretical results.

A Convergent Numerical Scheme for the Compressible Navier--Stokes Equations

In this paper, the three-dimensional compressible Navier--Stokes equations are considered on a periodic domain. We propose a semidiscrete numerical scheme and derive a priori bounds that ensure that the resulting system of ordinary differential equations (ODEs) is solvable for any $h>0$. An a posteriori observation that density remains uniformly bounded away from 0 will establish that a subsequence of the numerical solutions converges to a specific form of weak solution of the compressible Navier--Stokes equations.

Forced anisotropic mean curvature flow of graphs in relative geometry

The paper is concerned with the graph formulation of forced anisotropic mean curvature flow in the context of the heteroepitaxial growth of quantum dots. The problem is generalized by including anisotropy by means of Finsler metrics. A semi-discrete numerical scheme based on the method of lines is presented. Computational results with various anisotropy settings are shown and discussed.

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

AbstractWe show that a broad class of fully nonlinear, second‐order parabolic or elliptic PDEs can be realized as the Hamilton‐Jacobi‐Bellman equations of deterministic two‐person games. More precisely: given the PDE, we identify a deterministic, discrete‐time, two‐person game whose value function converges in the continuous‐time limit to the viscosity solution of the desired equation. Our game is, roughly speaking, a deterministic analogue of the stochastic representation recently introduced by Cheridito, Soner, Touzi, and Victoir. In the parabolic setting with no u‐dependence, it amounts to a semidiscrete numerical scheme whose timestep is a min‐max. Our result is interesting, because the usual control‐based interpretations of second‐order PDEs involve stochastic rather than deterministic control. © 2009 Wiley Periodicals, Inc.