A compact higher‐order ADI‐FDTD method

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition

ABSTRACTThis paper presents a novel one‐parameter linearized compact alternating direction implicit (ADI) scheme for solving the 2D and 3D non‐linear parabolic equations with distributed delay, where the ADI technique is used to reduce the computational cost greatly. The unique solvability, error estimates in and norms and stability of the proposed scheme are studied, and it is shown under some appropriate conditions that the scheme is convergent of order two in temporal direction and order four in spatial direction. Some numerical experiments are conducted to validate the theoretical results as well as the effectiveness and accuracy of the derived schemes.

A compact alternate direct implicit difference method for solving parabolic equation of multi-dimension

Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

On higher-order compact ADI schemes for the variable coefficient wave equation

Summary The higher order finite-difference operators complicate the easy implementation of boundary conditions and parallel computation. In this paper, we have derived an innovative explicit compact hybrid-grid finite-difference scheme, in which staggered grid and central grid are adopted together. Using this modified hybrid scheme it is possible to simulate the propagation of wave using a shorter finite-difference operator than published FD operators under the condition of not decreasing the accuracy or increasing computational cost. With shorter operator, it is beneficial to implement the boundary conditions and enhance the parallel computation efficiency. Then we indicate the feasibility of this hybrid operator through dispersion analysis and wave simulation. These analyses should be ideal for forward modeling and reverse time migration applications.

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

A sixth order finite difference scheme for the convection diffusion equation

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Péclet number 0 ≤ Pe ≤ 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.