An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.     

A Sequential Discontinuous Galerkin Method for the Coupling of Flow and Geomechanics

A decoupled and sequential numerical method is proposed and analyzed for solving the linear poroelasticity equations. Unlike other splitting approaches, this method is not iterative, which results in a speed-up of the computational time. The interior penalty discontinuous Galerkin method is employed for the spatial discretization and is combined with the backward Euler method for the time discretization. We provide a convergence analysis of the scheme along with numerical results that confirm the theoretical results.     

双曲型守恒律的一种三阶松弛格式

对一维双曲型守恒律,给出了一种形式更简单、计算量更小的三阶松弛格式.该格式以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础.由于不用求解Riemann问题和计算非线性通量函数的雅可比矩阵,所以本文格式保持了松弛格式简单的优点.数值试验表明:该方法具有较高的分辨率.     

A subgrid stabilization finite element method for incompressible magnetohydrodynamics

This paper studies a numerical scheme for approximating solutions of incompressible magnetohydrodynamic (MHD) equations that uses eddy viscosity stabilization only on the small scales of the fluid flow. This stabilization scheme for MHD equations uses a Galerkin finite element spatial discretization with Scott-Vogelius mixed finite elements and semi-implicit backward Euler temporal discretization. We prove its unconditional stability and prove how the coarse mesh can be chosen so that optimal convergence can be achieved. We also provide numerical experiments to confirm the theory and demonstrate the effectiveness of the scheme on a test problem for MHD channel flow.     

Symplectic spatial integration schemes for systems of balance equations

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.     

High order parameter-robust numerical method for time dependent singularly perturbed reaction–diffusion problems

We introduce a high order parameter-robust numerical method to solve a Dirichlet problem for one-dimensional time dependent singularly perturbed reaction-diffusion equation. A small parameter ε is multiplied with the second order spatial derivative in the equation. The parabolic boundary layers appear in the solution of the problem as the perturbation parameter ε tends to zero. To obtain the approximate solution of the problem we construct a numerical method by combining the Crank–Nicolson method on an uniform mesh in time direction, together with a hybrid scheme which is a suitable combination of a fourth order compact difference scheme and the standard central difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or ε-uniform in the sense that its numerical solution converges to the exact solution uniformly well with respect to the singular perturbation parameter ε. More specifically, we prove that the numerical method is uniformly convergent of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.     

Quasi-wavelet method for time-dependent fractional partial differential equation

In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential equations. The time variable is discretized by using the second order backward differentiation formula scheme, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. The comparisons of the present numerical results with the exact analytical solutions show that the quasi-wavelet method has distinctive local property and can achieve accurate results.     

Simulation of natural convection with the collocated clustered finite volume scheme

This paper presents numerical results obtained in the case of natural convection within non constant fluid density, using the collocated clustered finite volume (CCFV) scheme. The continuous equations are first given in a dimensionless form. Then we present the finite volume scheme with the principles and the spatial discretization used. Analytical tests illustrate the numerical behavior of this scheme according to the type of grid, of the pressure stabilization method and check the robustness of this scheme. Next, the results obtained on the square thermally driven cavity under large temperature differences show that the CCFV scheme accurately fits the reference results.     

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads to a large reduction of the computational scale. Under some realistic assumptions, the unconditional stability of the fully discrete scheme is proved. Moreover, the optimal error estimates are obtained, which are better than the existing results. Finally, some numerical results are given to verify the theoretical analysis. The difference between the motion of the Newton and non-Newton fluid is also observed.     

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.     

High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank-Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.     

Pricing American bond options using a penalty method

We develop a novel numerical method to price American options on a discount bond under the Cox–Ingrosll–Ross (CIR) model which is governed by a partial differential complementarity problem. We first propose a penalty approach to this complementarity problem, resulting in a nonlinear partial differential equation (PDE). To numerically solve this nonlinear PDE, we develop a novel fitted finite volume method for the spatial discretization, coupled with a fully implicit time-stepping scheme. We show that this full discretization scheme is consistent, stable and monotone, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the discretized nonlinear system, we design an iterative method and prove that the method is convergent. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of our methods.     

求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.     

A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains

A finite volume scheme which is based on fourth order accurate central differences in the spatial directions and on a hybrid explicit/semi-implicit time stepping scheme was developed to solve the incompressible Navier-Stokes equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity at the singularity of the cylindrical coordinate system and a new stability condition. The new method was applied in the direct numerical simulations (DNS) of the fully developed non-swirling turbulent flow through straight pipes with circular cross-section for the Reynolds number Re_τ = 360 based on the friction velocity u_τ and the pipe diameter. The obtained results are expressed in terms of statistical moments of the velocity components and are presented in comparison with those obtained with a second order accurate scheme and by measurements. It is shown that the fourth order spatial discretization leads to improved higher order statistical moments, while the first and the second order moments are more or less insensitive to the spatial discretization order.     

Radial-basis-function-based finite difference operator splitting method for pricing American options

We present a new radial-basis-function (RBF)-based numerical method for pricing European and American option problems. The governing equation is time semi-discretized by a linear-implicit backward difference method. The spatial discretization is done by using the RBF-based finite difference method. The numerical scheme first derived for an European option is extended for American options by using an operator splitting method. Numerical experiments with multiquadric RBF for one- and two-asset option problems are carried out, and the results obtained are compared with the existing ones.     

The characteristic-Galerkin method for advection-dominated problems—An assessment

In this note we present results of an accuracy analysis of a recent characteristic-based Galerkin method suited for advection-dominated problems. The analysis shows that the numerical propagation characteristics of the explicit time-stepping scheme which uses linear basis functions for spatial discretization are superior to those of the related classical Lax-Wendroff method and the implicit Crank-Nicolson scheme. The model is subjected to three analytical test problems which embrace many essential realistic features of environmental and coastal hydrodynamic applications: pure advection of a steep Gaussian profile, dispersion of a continuous source in an oscillating flow, and long-wave propagation with bottom frictional dissipation in a rectangular channel. The numerical results demonstrate that the accuracy achieved with the present scheme is excellent and comparable to that of a characteristic-based finite difference scheme which uses Hermitian cubic interpolating polynomials. The results reported herein suggest strongly further use and testing of this robust model in engineering practice.     

Distributed approximating functional approach to Burgers' equation in one and two space dimensions

The method of distributed approximating functionals (DAFs) is applied to Burgers' equation, as a typical nonlinear PDE, in one and two space dimensions. This equation is similar to, but simpler than, the Navier-Stokes equation in fluid dynamics. The present approach uses DAFs for spatial discretization and a Taylor expansion for time discretization. Several moderately large values of the Reynolds number are considered in the present application. The results obtained, which are in excellent agreement with the formally exact series solutions, are compared with those obtained by other authors using various different methods. It is found that a simple numerical propagation scheme based on DAFs provides highly accurate numerical solutions for Burgers' equation, while requiring very few grid points.     

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.