Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.     

A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates

In this paper, we extend our previous work (M.-C. Lai, A simple compact fourth-order Poisson solver on polar geometry, J. Comput. Phys. 182 (2002) 337–345) to 3D cases. More precisely, we present a spectral/finite difference scheme for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. Here the formal fourth-order accuracy means that the scheme is exactly fourth-order accurate while the poles are excluded and is third-order accurate otherwise. Despite the degradation of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condition is needed. The resulting linear system is then solved by the Bi-CGSTAB method with the preconditioner arising from the second-order discretization which shows the scalability with the problem size.     

Conservative finite difference schemes for the generalized Zakharov–Kuznetsov equations

This paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized Zakharov–Kuznetsov equations and the numerical solvability of the two-dimensional nonlinear wave equations. A finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized Zakharov–Kuznetsov equations hold. Our arguments are based on the procedure that D. Furihata has recently developed for real-valued nonlinear partial differential equations. Numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.     

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.     

Two difference schemes for the numerical solution of Maxwell’s equations as applied to extremely and super low frequency signal propagation in the Earth-ionosphere waveguide

Two explicit two-time-level difference schemes for the numerical solution of Maxwell’s equations are proposed to simulate propagation of small-amplitude extremely and super low frequency electromagnetic signals (200 Hz and lower) in the Earth-ionosphere waveguide with allowance for the tensor conductivity of the ionosphere. Both schemes rely on a new approach to time approximation, specifically, on Maxwell’s equations represented in integral form with respect to time. The spatial derivatives in both schemes are approximated to fourth-order accuracy. The first scheme uses field equations and is second-order accurate in time. The second scheme uses potential equations and is fourth-order accurate in time. Comparative test computations show that the schemes have a number of important advantages over those based on finite-difference approximations of time derivatives. Additionally, the potential scheme is shown to possess better properties than the field scheme.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

Fourth-order difference methods for the system of 2D nonlinear elliptic partial differential equations

We present the fourth-order finite difference methods for the system of 2D nonlinear elliptic equations using 9-grid points on a square region R subject to Dirichlet boundary conditions. The method has been tested on viscous, incompressible 2D Navier-Stokes equations. The numerical results show that the proposed methods produce accurate and oscillation-free solutions for large Reynolds numbers.     

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.     

Tracking entropy wave in ideal MHD equations by weighted ghost fluid method

The present paper is devoted to the computation of 1D magnetohydrodynamics (MHD) equations. We use a level set function to track the motion of the interface of the entropy wave. In addition, by considering a convex combination of the entropy-fixed ghost value and the density-fixed ghost value, the ghost fluid method (GFM) for multimaterial flows is extended. The resulting weighted ghost fluid method (WGFM) is combined with the fourth-order CWENO-type central-upwind scheme for approximating the numerical solution of 1D MHD equations in the real fluid and the ghost fluid. Computational results are eventually provided and discussed.     

Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation

We present a sixth-order explicit compact finite difference scheme to solve the three-dimensional (3D) convection-diffusion equation. We first use a multiscale multigrid method to solve the linear systems arising from a 19-point fourth-order discretization scheme to compute the fourth-order solutions on both a coarse grid and a fine grid. Then an operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid-independent convergence rate for solving convection-diffusion equations with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth-order compact (SOC) scheme, compared with the previously published fourth-order compact (FOC) scheme.     

An efficient transient Navier–Stokes solver on compact nonuniform space grids

In this paper, we propose an implicit higher-order compact (HOC) finite difference scheme for solving the two-dimensional (2D) unsteady Navier–Stokes (N–S) equations on nonuniform space grids. This temporally second-order accurate scheme which requires no transformation from the physical to the computational plane is at least third-order accurate in space, which has been demonstrated with numerical experiments. It efficiently captures both transient and steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. The proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation. Numerical results are presented and compared with analytical as well as established numerical data. Excellent comparison is obtained in all the cases.     

Numerical modeling of 1D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot model with frequency-independent coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space-time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.     

3D dynamic Green’s functions in a multilayered poroelastic half-space

The complete 3D dynamic Green’s functions in the multilayered poroelastic media are presented in this study. A method of potentials in cylindrical coordinate system is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to 3D wave propagation problems are obtained. After that, a three vector base and the propagator matrix method are introduced to treat 3D wave propagation problems in the stratified poroelastic half-space disturbed by buried sources. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. At last, the validity of the present approach for accurate and efficient calculating 3D dynamic Green’s functions of a multilayered poroelastic half-space is confirmed by comparing the numerical results with the known exact analytical solutions of a uniform poroelastic half-space.     

A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems

In this article, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank–Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection–diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection–diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods. The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.     

Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems

The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto–Sivashinsky equation and extended Fisher–Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.     

Galerkin-Chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations

In this paper, we present a high-order accurate method for two-dimensional semilinear parabolic equations. The method is based on a Galerkin-Chebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Optimal a priori error bound is derived in the weighted \(L^{2}_{\omega }\)-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence properties of the method.     

Maximum norm error estimates of efficient difference schemes for second-order wave equations

The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results.