Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Numerical integration of the time-dependent equations of motion for taylor vortex flow

A method is presented for the finite difference solution of the equations of fluid motion. The complete Navier-Stokes equations are expressed in terms of tangential velocity, vorticity and stream function. The transformed equations are solved using an alternating direction implicit scheme. The classical problem of hydrodynamic stability of the rotational Couette flow is solved in two dimensions. Comparison with other numerical and experimental works shows that the method reported here is computationally stable, even when used with coarse grids and relatively large time increments.     

A FV-TD electromagnetic solver using adaptive Cartesian grids

A second-order finite-volume (FV) method has been developed to solve the time-domain (TD) Maxwell equations, which govern the dynamics of electromagnetic waves. The computational electromagnetic (CEM) solver is capable of handling arbitrary grids, including structured, unstructured, and adaptive Cartesian grids, which are topologically arbitrary. It is argued in this paper that the adaptive Cartesian grid is better than a tetrahedral grid for complex geometries considering both efficiency and accuracy. A cell-wise linear reconstruction scheme is employed to achieve second-order spatial accuracy. Second-order time accuracy is obtained through a two-step Runge-Kutta scheme. Issues on automatic adaptive Cartesian grid generation such as cell-cutting and cell-merging are discussed. A multi-dimensional characteristic absorbing boundary condition (MDC-ABC) is developed at the truncated far-field boundary to reduce reflected waves from this artificial boundary. The CEM solver is demonstrated with several test cases with analytical solutions.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

Non-hydrostatic finite volume model for non-linear waves interacting with structures

An efficient non-hydrostatic finite volume model is developed and applied to simulate non-linear waves interacting with structures. The unsteady Navier–Stokes equations are solved in a 3D grid made of polyhedrons, which are built from a 2D horizontal unstructured grid by adding several horizontal layers. A new grid arrangement in the vertical direction is proposed, which renders the resulting model is relatively simple. Moreover, the discretized Poisson equation for pressure is symmetric and positive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method. Several test cases including solitary wave interacting with a submerged structure, solitary wave scattering from a vertical circular cylinder and an array of four circular cylinders are used to demonstrate the capability of the model on simulating non-linear waves interacting with structures. In all cases, the model gives satisfactory results in comparison with analytical solutions, experimental data and other published numerical results.     

Finite difference method and a spurious reflection of waves in a layer of fluid

For a layer of fluid and boundary conditions considered in this paper analytical solutions of the Helmholtz equation exist and are well known. When the continuum is replaced by a discrete space of chosen points with a regular grid, analogous solutions may be calculated. In the case of two uniform grids with sudden mesh size change from one grid to the other, in the layer a deformation of discrete solutions occurs. The deformation assumes the form of a spurious wave reflected from the boundary of the two grids. Moreover, the transmitted wave amplitude is changed. If the wave propagates from a grid of smaller mesh size, then the transmitted wave has a larger amplitude than the incident wave. The explicit formulae obtained enable us to calculate the degree of deformation of discrete solutions for the finite difference method and the nonuniform grid.     

A numerical investigation of errors arising in applying the galerkin method of the solution of nonlinear partial differential equations

The Galerkin method is applied to the solution of Burgers' equation and a nonlinear wave equation, using expansions of B-splines of increasing number of terms and order of spline. The accuracy of the solutions obtained numerically is compared with analytical solutions, and the effect upon accuracy of increasing the order of spline and number of terms in the expansion is considered for a variety of initial conditions corresponding to waves having a range of wavelengths.Burgers' equation is used as a model for the hydrodynamic shallow water equations, and results illustrate the importance of using a sufficient number of functions in the expansion to accurately model the distortion of a wave progressing into a shallow water region where shorter waves contribute appreciably to the total wave profile.     

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.     

Variable grid scheme for discontinuous grid spacing and derivatives

A Crank-Nicolson type finite-difference scheme is developed for solving boundary layer flows on arbitrary grids and with jumps in viscosity and density. The method is applied to the similar equations and two approaches are obtained depending upon the linearization of terms. One of these approaches can be developed from the box scheme formulation. In some cases, difference relations for derivatives are those obtained in the variable grid scheme developed previously. Numerical solution verify that the difference techniques have second-order behavior as the grid system is refined. A wall velocity gradient relation is determined which gives second-order accuracy for all grids considered.     

A Cascadic Multigrid Algorithm for Semilinear Indefinite Elliptic Problems

We propose a cascadic multigrid algorithm for a semilinear indefinite elliptic problem. We use a standard finite element discretization with piecewise linear finite elements. The arising nonlinear equations are solved by a cascadic organization of Newton's method with frozen derivative on a sequence of nested grids. This gives a simple version of a multigrid method without projections on coarser grids. The cascadic multigrid algorithm starts on a comparatively coarse grid where the number of unknowns is small enough to obtain an approximate solution within sufficiently high precision without substantial computational effort. On each finer grid we perform exactly one Newton step taking the approximate solution from the coarsest grid as initial guess. The linear Newton systems are solved iteratively by a Jacobi-type iteration with special parameters using the approximate solution from the previous grid as initial guess. We prove that for a sufficiently fine initial grid and for a sufficiently good start approximation the algorithm yields an approximate solution within the discretization error on the finest grid and that the method has multigrid complexity with logarithmic multiplier. Received February 1999, revised July 13, 1999     

Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton–Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.     

Accelerated discrete velocity method for axial-symmetric flows of gaseous mixtures as defined by the McCormack kinetic model

An accelerated discrete velocity method is presented to calculate the steady axial-symmetric flows of gaseous mixtures defined by the McCormack kinetic model. The scheme is formulated in cylindrical coordinates. Diffusion equations for the macroscopic velocity and the heat-flow are derived on the basis of the projected McCormack equations. The solutions of the kinetic equations are carried out iteratively by using the discrete velocity method. The diffusion equations are also solved in each stage of the iteration in order to accelerate the scheme. Pressure driven flows of He/Xe and Ne/Ar mixtures through a cylindrical tube are simulated in order to study the computational performance of the approach. It is shown that the required number of iterations and the computational times are significantly reduced at intermediate and large values of the rarefaction parameter by using the accelerated method. In the hydrodynamic limit, the flow rates of the components converge to the corresponding slip flow results. Flows driven by mole fraction gradient are also successfully simulated. Typical velocity and heat-flow profiles for pressure driven flow of He/Xe mixture are shown and commented on.     

A front-tracking method with projected interface conditions for compressible multi-fluid flows

A front-tracking method for compressible multi-fluid flows is presented, where marker points are used both for tracking fluid interfaces and also for constructing the Riemann problem on the interfaces. The Riemann problem between the two fluid phases (defined in the interface normal direction) is solved using the exact Riemann solver on the marker points. The solutions are projected onto fixed grid points and then extrapolated into the corresponding ghost-fluid regions, to be used as boundary conditions. Each fluid phase is solved separately as in the ghost-fluid method. The proposed procedures, especially the projection of the exact Riemann solutions onto the fluid grids, are designed to be simple and consistent in any spatial dimensions. Several multi-fluid problems, including the breakup of a water cylinder induced by the passage of a shock wave were computed in order to demonstrate the capability of the proposed method.     

The unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models: Revisited

The unconditionally positive-convergent implicit scheme for two-equation turbulence models, originally developed by Mor-Yossef and Levy, is revisited. A compact, simple, and uniform reformulation of the method for the use of both structured and unstructured grid based flow solvers is presented. An analytical proof of the scheme revision is given showing that positivity of the turbulence model solutions and convergence of the turbulence model equations are guaranteed for any time step. Numerical experiments are conducted, simulating two test cases of three-dimensional complex flow fields using structured and hybrid unstructured grids. To demonstrate the overall scheme’s robustness, it is applied to non-linear k-ω and non-linear k- turbulence models. Results from the numerical simulations show that the scheme exhibits very good convergence characteristics, is robust, and it always preserves the positivity of the turbulence model dependent variables, even for an infinite time step.     

Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves

Considered here are Boussinesq systems of equations of surface water wave theory over a variable bottom. A simplified such Boussinesq system is derived and solved numerically by the standard Galerkin-finite element method. We study by numerical means the generation of tsunami waves due to bottom deformation and we compare the results with analytical solutions of the linearized Euler equations. Moreover, we study tsunami wave propagation in the case of the Java 2006 event, comparing the results of the Boussinesq model with those produced by the finite-difference code MOST, that solves the shallow water wave equations.     

The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

An abstract framework ofauxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a furthernested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.     

Implicit finite difference techniques for the advection–diffusion equation using spreadsheets

This study proposes one-dimensional advection–diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implementation, solutions are implicitly obtained for the BTCS, Upwind and Crank–Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. Thus, it is not required a solution of simultaneous equations for each time step using matrix algebra. Two examples which, have the numerical and analytical solutions in literature, are solved in order to test the ADEISS performance. Both examples are solved for three schemes. It has been determined that the Crank–Nicolson scheme is in good agreement with the analytical solution; however the results of the BTCS and the Upwind schemes are lower than the analytical solution. The Upwind scheme suffers from considerably numerical diffusion whereas the BTCS scheme does not produce numerical diffusion. Thus, it provided better results than Upwind scheme which are closer to analytical results depending on the selected parameters. The ADEISS implementation is a computationally convenient procedure for the three well-known methods in the literature: The BTCS, Upwind and Crank–Nicolson.     

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.     

Auxiliary equation method for time-fractional differential equations with conformable derivative

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.