Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation

In this paper, two efficient fourth-order compact finite difference algorithms have been developed to solve the one-dimensional Burgers’ equation: u _t +u u _x =ε u _xx . The methods are based on the Hopf–Cole transformation, Richardson's extrapolation, and multilevel grids. In both methods, we first transform the original nonlinear Burgers’ equation into a linear heat equation: w _t =ε w _xx using the Hopf–Cole transformation, which is given as u=?2ε (w _x /w). In the first method, the resulted heat equation is solved by the second-order accurate Crank–Nicholson algorithm while w _x is approximated by central finite difference, which is also second-order accurate. Richardson's extrapolation technique is then applied in both time and space to obtain fourth-order accuracy. In the second method, to reduce the cancellation error in approximating w _x , we derive the heat equation satisfied by w _x , which is then solved by the Crank–Nicholson algorithm. The original Dirichlet boundary condition is transformed into the Robin boundary condition, which is also approximated using second-order central finite difference. Finally, Richardson's extrapolation and multilevel grid techniques are applied in both time and space to obtain fourth-order accuracy. To study the efficiency, accuracy and robustness, we solved two numerical examples and the results are compared with those of two other higher-order methods proposed in W. Liao [An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206(2) (2008), pp. 755–764] and I.A. Hassanien, A.A. Salama, and H.A. Hosham [Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput. 170 (2005), pp. 781–800].     

Time-local formulation and identification of implicit Volterra models by use of diffusive representation

We present a time-continuous identification method for nonlinear dynamic Volterra models of the form HX=f(u,X)+v with H, a causal convolution operator. It is mainly based on a suitable parameterization of H deduced from the so-called diffusive representation, which is devoted to state representations of integral operators. Following this approach, the complex dynamic nature of H can be summarized by a few numerical parameters on which the identification of the dynamic part of the model will focus. The method is validated on a physical numerical example.     

A novel numerical method for a class of problems with the transition layer and Burgers’ equation

A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ? u _xx +a(u+f(x))u _x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ? u _xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.     

Eigenvalue analysis of linearized error dynamics of measurement integrated flow simulation

Experimental measurement and numerical simulation are the typical methods employed in flow analysis. Both methods have advantages and disadvantages and it is difficult to correctly reproduce real flows with inherent uncertainties. In order to overcome this problem, measurement-integrated (MI) simulation has been proposed in which measurement and simulation are integrated based on the observer theory. The validity of MI simulation has been proved in several applications. However the feedback law critical in MI simulation has been designed by trial and error based on physical considerations. Development of a general theory for the design of MI simulation is critical for its widespread use. In this study, as a fundamental consideration to construct a general theory of MI simulation, we formulated a linearized error dynamics equation to express time development of the error between the simulation and the real flow, and an equation for eigenvalue analysis. Primary advantage of the proposed method is to provide a framework to design a feedback gain of MI simulation based on a standard linear dynamical system theory. The validity of the method was investigated by comparison of the results of the eigenvalue analysis and those of the numerical experiment for the low-order model problem of the turbulent flow in a square duct with various feedback gains in the case of feedback with all velocity components and two velocity components. From the eigenvalue analysis in the case without feedback, the error dynamics was unstable and the error increased exponentially. When the feedback gain k_u > 0.98 with feedback of all velocity components or k_u > 1.67 with feedback of u₁, u₂ velocity components, all eigenvalues were stable. In the numerical experiment, the critical feedback gains obtained from eigenvalue analysis quantitatively agreed with the lower limit of the feedback gain to reduce the steady error in MI simulation. In the comparison of the time constant for the reduction of the error norm, the time constant obtained from the eigenvalue analysis agreed with those from the numerical experiment. The eigenvalue analysis of the linearized error dynamics formulated in this study was effective in evaluation of the effect of the feedback gain of the MI simulation.     

A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.     

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det (D ² u ⁰)=f (>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74–98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampère equation, submitted). In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −εΔ² u ^ε +det D ² u ^ε =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ^ε of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u^e-u^e_hu^{\varepsilon}-u^{\varepsilon}_{h}. Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u⁰-u_h^eu^{0}-u_{h}^{\varepsilon}, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.     

Totally Discrete Explicit and Semi-implicit Euler Methods for a Blow-up Problem in Several Space Dimensions

The equation u _t =Δu+u ^p with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.     

A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u₀(x) and homogeneous Dirichlet boundary conditions in a bounded interval [0, L]. We study a semidiscrete approximation scheme based on the pseudo-spectral method on Chebyshev–Gauss–Lobatto nodes. In order to preserve the high accuracy of the spectral approximation we use an approach based on the evaluation of the Mittag-Leffler function on matrix arguments for the integration along the time variable. Some examples are presented and numerical experiments illustrate the effectiveness of the proposed approach.     

Keller's Box-Scheme for the One-Dimensional Stationary Convection-Diffusion Equation

u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu _x −ɛu _xx =f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method. Received June 7, 2001; revised October 2, 2001     

H 1-Galerkin Mixed Finite Element Method for the Regularized Long Wave Equation

In this paper, an H¹-Galerkin mixed finite element method is proposed for the 1-D regularized long wave (RLW) equation u_t+u_x+uu_x−δu_xxt=0. The existence of unique solutions of the semi-discrete and fully discrete H¹-Galerkin mixed finite element methods is proved, and optimal error estimates are established. Our method can simultaneously approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.     

Solving the forced Duffing equation with integral boundary conditions in the reproducing kernel space

In this paper, we propose a new method to solve the forced Duffing equation with integral boundary conditions. Its exact solution is represented in the form of a series in the reproducing kernel space. The n-term approximation u _n (x) of the exact solution u(x) is proved to converge to the exact solution. Some numerical examples are displayed to demonstrate the accuracy of the present method.     

Strong Stability Preserving Properties of Runge–Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation u _t =Lu where L is a linear operator. We present optimal SSP Runge–Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge–Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.     

Numerical Shape-From-Shading for Discontinuous Photographic Images

The height, u(x, y), of a continuous, Lambertian surface of known albedo (i.e., grayness) is related to u(x, y), information recoverable from a black and white flash photograph of the surface, by the partial differential equation

Extended one-step time-integration schemes for convection-diffusion equations

We first describe a one-parameter family of unconditionally stable third-order time-integration schemes for the convection-diffusion equation: u_t + cu_x = vu_xx, based on the extended trapezoidal formulas of Usmani and Agarwal [1]. Interestingly, there exists a method that is fourth order as well as unconditionally stable. We then describe a one-parameter family of unconditionally stable fourth-order time-integration schemes, based on the extended Simpson rules of Chawla et al. [2]. Again, there exists a method that is fifth order as well as unconditionally stable. The stability and accuracy of the obtained methods are tested, and compared with the widely used method of Crank-Nicolson, by considering three problems of practical interest.     

$$L_2$$ Stability of Explicit Runge–Kutta Schemes

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.     

An accurate algorithm for computing the eigenvalues of a polygonal membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω₁, Ω₂,...; on each $\text{Ω}{}_{\mathrm{i}}$, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.     

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Fast algorithms for high-order numerical methods for space-fractional diffusion equations

In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant (CS) representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis.     

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x _n+j , i = 0(1)k and at an off grid point x = x _n+u , where k is the step number of the method and u is an arbitrary rational number in (x _n , x _n+k ). A predictor of order 2k ? 1 is also proposed to cater for y _n+k in the main method. Taylor series expansion is employed for the calculation of y _n+1, y _n+2, y _n+u and their higher derivatives. Evaluation of the resulting method at x = x _n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.