• Abstract

In this paper, we prove the optimal error estimates in $L^2$ norm of the semi-discrete local discontinuous Galerkin (LDG) method for the thin film epitaxy problem without slope selection. To relax the severe time step restriction of explicit time marching methods, we employ a class of exponential time differencing (ETD) schemes for time integration, which is based on a linear convex splitting principle. Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.

• Abstract

The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem, due to the global properties of fractional differential operators. In this paper, we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable. By the proposed first-order optimality condition consisting of a Lagrange multiplier, we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations. Furthermore, a priori error estimates for state, adjoint state and control variables are discussed in details. Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.

• Abstract

In this paper, we propose, analyze and numerically validate a conservative finite element method for the nonlinear Schrödinger equation. A scalar auxiliary variable (SAV) is introduced to reformulate the nonlinear Schrödinger equation into an equivalent system and to transform the energy into a quadratic form. We use the standard continuous finite element method for the spatial discretization, and the relaxation Runge-Kutta method for the time discretization. Both mass and energy conservation laws are shown for the semi-discrete finite element scheme, and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method. Numerical examples are presented to demonstrate the accuracy of the proposed method, and the conservation of mass and energy in long time simulations.

• Abstract

In this paper, we consider the energy conserving numerical scheme for coupled nonlinear Klein-Gordon equations. We propose energy conserving finite element method and get the unconditional superconvergence result $\mathcal{O}(h^2+∆t^2 )$ by using the error splitting technique and postprocessing interppolation. Numerical experiments are carried out to support our theoretical results.

• Abstract

This paper numerically and analytically investigates a non-linear static, two-dimensional thermoelastic analysis in the radial and tangential directions of a cylindrical shell made of functionally graded materials. The dependence of material properties on temperature makes the heat governing equations non-linear. To obtain the temperature field analytically, the heat conduction equation is linearized and exactly solved using a linearizing transformation, then this exact solution is substituted in the Lamme-Navier equations, and the elasticity equations are numerically solved using a second-order central finite difference method, and displacement and stress distributions are obtained. Finally, the temperature field, stress, and displacement distributions are presented, and the effect of inhomogeneous parameters on them is examined and discussed. The correctness and accuracy of the exact analytical solution of the temperature field are illustrated by a comparison with the numerical solution. The results show a good agreement.

• Abstract

In this paper, numerical methods for the time-changed stochastic differential equations of the form $$d\Upsilon(t)=a(\Upsilon(t))dt+b(\Upsilon(t))dE(t)+σ(\Upsilon(t))dB(E(t))$$ are investigated, where all the coefficients $a(·),$ $b(·)$ and $σ(·)$ are allowed to contain some super-linearly growing terms. An explicit method is proposed by using the idea of truncating terms that grow too fast. Strong convergence in the finite time of the proposed method is proved and the convergence rate is obtained. The proposed method is also proved to be able to reproduce the asymptotic stability of the underlying equation in the almost sure sense. Simulations are provided to demonstrate the theoretical results.

• Abstract

In this paper, the natural convection of a complex fluid that contains both nanoparticles and gyrotactic microorganisms in a heated square cavity is considered. The Buongiorno model is applied to descirbe the nanofluid behaviours. Both the top and bottom horizontal walls of the cavity are adiabatic, and there is a temperature difference between the left and right vertical walls. The non-dimensional governing equations are obtained when the stream-vorticity formulation of function is used, which are solved by the recently developed robust Coiflet wavelet homotopy analysis method. A rigid verification for the solver is given. Besides, the effects of various physics parameters including the Rayleigh number, the buoyancy ratio parameter, the bioconvection Rayleigh number, the Prandtl number, the Brownian motion parameter, the thermophoresis parameter, the heat generation parameter, the Lewis number, the bioconvection Peclet number and the Schmidt number on this complicated natural convection are examined. It is known that natural convection is closely related to our daily life owing to its wide existence in nature and engineering applications. We believe that our work will make a significant contribution to a better understanding of the natural convection of a complex fluid in a cavity with suspensions of both inorganic nanoparticles and organic microorganisms.

• Abstract

Localized point sources (monopoles) in an acoustical domain are implemented to a three dimensional non-singular Helmholtz boundary element method in the frequency domain. It allows for the straightforward use of higher order surface elements on the boundaries of the problem. It will been shown that the effect of the monopole sources ends up on the right hand side of the resulting matrix system. Some carefully selected examples are studied, such as point sources near and within a concentric spherical core-shell scatterer (with theoretical verification), near a curved focusing surface and near a multi-scale and multi-domain acoustic lens.

• Abstract

In this work, by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations, we will propose new fully discrete multistep schemes called "Sinc-multistep schemes" for forward backward stochastic differential equations (FBSDEs). The schemes avoid spatial interpolations and admit high order of convergence. The stability and the $K$-th order error estimates in time for the $K$-step Sinc multistep schemes are theoretically proved $(1≤K≤6).$ This seems to be the first time for analyzing fully time-space discrete multistep schemes for FBSDEs. Numerical examples are also presented to demonstrate the effectiveness, stability, and high order of convergence of the proposed schemes.

• Abstract

The Ahmed model is a standard bluff body used to study the flow behavior around an automobile. An important issue when investigating turbulent flow fields is the large computational load driven by accurate prediction approaches, such as the large eddy simulation model. In this paper, we present a powerful domain decomposition method-based parallel solver to efficiently utilize existing supercomputer resources. The 3D unsteady incompressible Navier–Stokes equations with a subgridscale (SGS) fluid model are discretized on a pure unstructured tetrahedral grid by a stable $P_1−P_1$ finite element method in space, while an implicit second-order backward differentiation formula is employed for the time discretization. We then solve the nonlinear algebraic system by means of the Newton–Krylov–Schwarz method by imposing a restricted additive Schwarz (RAS) right preconditioner for the parallel setting. We validate the proposed method toward the comparison of the flow field, including the velocity profiles and flow structures, with experimental investigations, and we show the parallel efficiency and scalability of the solver with up to 8192 processors.

• Abstract

This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schrödinger equation in two dimensions, which is not restricted that the nonlinear term is mere cubic. The new framework of convergence analysis consists of two steps. In the first step, by truncating the nonlinear term into a global Lipschitz function, an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete $L^2$ norm; followed in the second step, the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique, as implies that the two numerical methods are the same one. Under our framework of convergence analysis, with neither any restriction on the grid ratio nor any requirement of the small initial value, we establish the error estimate of the proposed conservative Fourier pseudo-spectral method, while previous work requires the certain restriction for the focusing case. The error bound is proved to be of $\mathcal{O}(h^r+τ^2)$ with grid size $h$ and time step $τ.$ In fact, the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schrödinger-type equations. Numerical results are conducted to indicate the accuracy and efficiency of the proposed method, and investigate the effect of the nonlinear term and initial data on the blow-up solution.

• Abstract

In this paper, we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations (VFS-DEs). We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method. In particular, when the fractional order is no longer varying, the conclusions obtained are consistent with the relevant conclusions in the existing literature. Finally, the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

• Abstract

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational mechanics. During the past few decades, the method of fundamental solutions (MFS) has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering applications. The classical MFS formulation, however, leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering simulations. In this paper, a localized version of the MFS (LMFS) is devised for electroelastic analysis of two-dimensional (2D) piezoelectric structures. In the LMFS, the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square (MLS) technique are employed. Different to the classical MFS, the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale simulations. Preliminary numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

• Abstract

In this work, we study the coercivity of a family of quadratic finite volume element (FVE) schemes over triangular meshes for solving elliptic boundary value problems. The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as $H^1$ and $L^2$ error estimates. The novelty of this paper is that, each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. As a result, we reach a sufficient condition to guarantee the existence, uniqueness and coercivity result of the FVE solution on general triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic and computable expressions are obtained. By comparison, the existing minimum angle conditions were obtained numerically from a computer program. Theoretical findings are conformed with the numerical results.

• Abstract

A unified solution framework is proposed for efficiently solving conjugate fluid and solid heat transfer problems. The unified solution is solely governed by the compressible Navier-Stokes (N-S) equations in both fluid and solid domains. Such method not only provides the computational capability for solid heat transfer simulations with existing successful N-S flow solvers, but also can relax time-stepping restrictions often imposed by the interface conditions for conjugate fluid and solid heat transfer. This paper serves as Part I of the proposed unified solution framework and addresses the handling of solid heat conduction with the nondimensional N-S equations. Specially, a parallel, adaptive high-order discontinuous Galerkin unified solver has been developed and applied to solve solid heat transfer problems under various boundary conditions.

• Abstract

In this paper, based on the imaginary time gradient flow model in the density functional theory, a scalar auxiliary variable (SAV) method is developed for the ground state calculation of a given electronic structure system. To handle the orthonormality constraint on those wave functions, two kinds of penalty terms are introduced in designing the modified energy functional in SAV, i.e., one for the norm preserving of each wave function, another for the orthogonality between each pair of different wave functions. A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization. Theoretically, the desired unconditional decay of modified energy can be obtained from our method, while computationally, both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments. More importantly, numerical results show that the orthonormality among those wave functions can be automatically preserved, without explicitly preserving orthogonalization operations. This implies the potential of our method in large-scale simulations in density functional theory.

• Abstract

A partial Runge-Kutta Discontinuous Galerkin (RKDG) method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations written in semi-Lagrangian formulation on moving quadrilateral meshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG method as our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD. Since the divergence of the magnetic field in 2D is independent of its $z$-direction component, an exactly divergence-free numerical magnetic field can be obtained by this treatment. We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy, non-oscillatory property and preservation of the exactly divergence-free property of our method.

• Abstract

The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time. The Korteweg-de Vries (KdV) equation, the Burgers equation and the Korteweg-de Vries-Burgers (KdVB) equation are examined as illustrative examples. Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm. Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies. Furthermore, it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter. It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems. It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.

• Abstract

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

• Abstract

In this article, a robust, effective, and scale-invariant weighted compact nonlinear scheme (WCNS) is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes. The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically, which is caused by a loss of sub-stencils’ adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor. A new nonlinear weight is devised by using an average of the function values and the descaling function, providing the new WCNS schemes (WCNS-Zm/Dm) with many attractive properties. The ENO-property, Si-property and Cp-property of the new WCNS schemes are validated numerically. Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property, while only the WCNS-Dm scheme satisfies the Cp-property. In addition, the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved. Several one-dimensional shock tube problems, and two-dimensional double Mach reflection (DMR) problem and the Riemann IVP problem are simulated to illustrate the ENO-property and Si-property of the scale-invariant WCNS-Zm/Dm schemes.