Optimal control computation for integro-differential aerodynamic equations

In order to maintain spectrally accurate solutions, the grids on which a non-linear physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a pseudospectral computational method of solving integro-differential systems with quadratic performance index. The proposed method is based on the idea of relating grid points to the structure of orthogonal interpolating polynomials. The optimal control and the trajectory are approximated by the m th degree interpolating polynomial. This interpolating polynomial is spectrally constructed using Legendre–Gauss–Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. The integrals involved in the formulation of the problem are calculated by Gauss–Lobatto integration rule, thereby reducing the problem to a mathematical programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. An illustrative example is included to confirm the convergence of the pseudospectral Legendre method, and a comparison is made with an existing result in the literature. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A new type of boundary treatment for wave propagation

Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be \(L^2\)-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

High resolution multi-moment finite volume method for supersonic combustion on unstructured grids

In this study, we present a novel numerical model for simulating detonation waves on unstructured grids. In contrast to the conventional finite volume method (FVM), two types of moment comprising the volume-integrated average (VIA) and the point value (PV) at the cell vertex are treated as the evolution variables for the reacting Euler equations. The VIA is computed based on a finite volume formulation of the flux form where the conventional Riemann problem is solved by the HLLC Riemann solver. The PV is updated in a point-wise manner by using the differential formulation where the Roe solver is used to compute the differential Riemann problems. In order to increase the accuracy around discontinuities, numerical oscillations and dissipations are reduced using the boundary variation diminishing algorithm. Convergence tests demonstrated that the proposed model could achieve third-order accuracy with unstructured grids for reacting Euler equations. The high resolution property of the proposed method was verified based on simulations of several detonation wave propagation problems in two and three dimensions. In particular, the current model could resolve the cellular structures with fewer degrees of freedom for the unstable oblique detonation wave problem. These fine structures may be smoothed out by the conventional FVM due to the excessive amount of numerical dissipation errors. Importantly, a simulation of stiff detonation waves showed that the proposed method could capture the correct position of the reaction front whereas the conventional FVMs produced spurious phenomena. Thus, the proposed model can obtain highly accurate solutions for detonation problems on unstructured grids, which is highly advantageous for real applications involving complex geometrical configurations.     

Convergence of the fixed pivot technique for continuous Smoluchowski coagulation equation

In this work, the fixed pivot technique (FPT) [2] is analyzed for nonlinear continuous Smoluchowski coagulation equation on four different types of grids. More importantly, the FPT gives the accuracy of second order for uniform and geometric grids while it reduces the order of accuracy by one on a locally uniform grid. At the end, the scheme is unfortunately zero order accurate on random grids. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

A kinetic energy preserving DG scheme applied to mechanical FSI

In the context of mechanical fluid-structure interaction (FSI) comprising moving or deforming structures, fluid discretizations need to cope with time-dependent fluid domains and resulting grid deformations in addition to the general challenges regarding e.g. boundary layers and turbulent phenomena. Recent approaches in the simulation of compressible turbulent flow are based on so-called split forms of conservation laws to guarantee the preservation of secondary physical quantities such as kinetic energy. For the simulation of turbulent flows, this often leads to a better representation of the kinetic energy spectrum. Initially, kinetic energy preserving(KEP) DG schemes have been constructed on Gauss-Legendre-Lobatto(GLL) nodes containing the interval end points, however, KEP DG schemes based on the classical Gauss-Legendre(GL) nodes are potentially more accurate and may be also more efficient than its GLL variant for certain applications. In this work, the KEP-DG schemes both on GL and GLL nodes are applied to the classical moving piston test case via an ALE formulation on moving fluid grids showing a more accurate frequency representation of the structure displacement in case of GLL nodes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of non‐stationary aero‐autoelasticity integro‐differential operator equations

The spectral method of G. N. Elnagar, which yields spectral convergence rate for the approximate solutions of Fredholm and Volterra–Hammerstein integral equations, is generalized in order to solve the larger class of integro‐differential functional operator equations with spectral accuracy. In order to obtain spectrally accurate solutions, the grids on which the above class of problems is to be solved must also be obtained by spectrally accurate techniques. The proposed method is based on the idea of relating, spectrally constructed, grid points to the structure of projection operators which will be used to approximate the control vector and the associated state vector. These projection operators are spectrally constructed using Chebyshev–Gauss–Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. Simulation studies demonstrate computational advantages relative to other methods in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015     

On richardson extrapolation for finite difference methods on regular grids

Summary Difference solutions of partial differential equations can in certain cases be expanded by even powers of a discretization parameterh. If we haven solutions corresponding to different mesh widthsh ₁,...,h _n we can improve the accuracy by Richardson extrapolation and get a solution of order 2n, yet only on the intersection of all grids used, i.e. normally on the coarsest grid. To interpolate this high order solution with the same accuracy in points not belonging to all grids, we need 2n points in an interval of length (2n–1)h ₁.This drawback can be avoided by combining such an interpolation with the extrapolation byh. In this case the approximation depends only on grid points in an interval of length 3/2h ₁. The length of this interval is independent of the desired order.By combining this approach with the method of Kreiss, boundary conditions on curved boundaries can be discretized with a high order even on coarse grids.This paper is based on a lecture held at the 5th Sanmarinian University Session of the International Academy of Sciences San Marino, at San Marino, 1988-08-27-1988-09-05     

An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

In this paper, we present two families of second-order and third-order explicit methods for numerical integration of initial-value problems of ordinary differential equations. Firstly, a family of second-order methods with two free parameters is derived by considering a suitable rational approximation to the theoretical solution of the problem at some grid points. Imposing that the principal term of the local truncation error of this family vanishes, we obtain an expression for one of the parameters in terms of the other. With this approach, a new one-parameter family of third-order methods is obtained. By selecting any 3(2) pair of second and third order methods, they can be implemented as an embedded type method, thus leading to a variable step-size formulation. We have considered one 3(2) pair of second and third order methods and made a comparison of numerical results with several ode solvers which are currently used in practice. The comparison of numerical results shows that the embedded 3(2) pair outperforms the methods considered for comparison.     

Models and solution techniques for production planning problems with increasing byproducts

We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and we derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain. We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.     

Central schemes for Ideal magnetohydrodynamics

We present second-order accurate central finite volume methods adapted to three-dimensional ideal magnetohydrodynamics problems. These methods alternate between two staggered grids, thus leading to Riemann solver-free algorithms with relatively favorable computing times. The div· B = 0 constraint on the magnetic field is enforced with a suitable adaptation of the constrained transport method to our central schemes. Numerical experiments show the feasibility of the proposed methods and our results are in good agreement with existing results in the recent literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme

Computational fluid dynamics (CFD) has become increasingly used in the industry for the simulation of flows. Nevertheless, the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult. From this point of view, the development of higher-order methods for unstructured grids is desirable. The finite volume method can be used with unstructured grids, but unfortunately it is difficult to achieve an order of accuracy higher than two, and the common approach is a simple extension of the one-dimensional case. The increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives. This problem is overcome with the application of the Moving Least Squares (MLS) technique on a finite volume framework. In this work we present the application of this method (FV-MLS) to the solution of aeroacoustic problems.     

Higher order numerical discretizations for exterior and biharmonic type PDEs

A higher order numerical discretization technique based on Minimum Sobolev Norm (MSN) interpolation was introduced in our previous work. In this article, the discretization technique is presented as a tool to solve two hard classes of PDEs, namely, the exterior Laplace problem and the biharmonic problem. The exterior Laplace problem is compactified and the resultant near singular PDE is solved using this technique. This finite difference type method is then used to discretize and solve biharmonic type PDEs. A simple book keeping trick of using Ghost points is used to obtain a perfectly constrained discrete system. Numerical results such as discretization error, condition number estimate, and solution error are presented. For both classes of PDEs, variable coefficient examples on complicated geometries and irregular grids are considered. The method is seen to have high order of convergence in all these cases through numerical evidence. Perhaps for the first time, such a systematic higher order procedure for irregular grids and variable coefficient cases is now available. Though not discussed in the paper, the idea seems to be easily generalizable to finite element type techniques as well.     

A Large Time Step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws: Application to the homogeneous shallow water equations

In this work, a Large Time Step (LTS) explicit finite volume scheme designed to allow CFL > 1 is applied to the numerical resolution of 2D scalar and systems of conservation laws on triangular grids. Based on the flux difference splitting formulation, a special concern is put on finding the way of packing the information to compute the numerical solution when working on unstructured grids. Not only the cell areas but also the length of the interfaces and their orientation are questions of interest to send the information from each edge or interface. The information to update the cell variables is computed according to the local average discrete velocity and the orientation of the edges of the cells involved. The performance of these ideas is tested and compared with the conventional explicit first order and second order schemes in academic configurations for the 2D linear scalar equation and for 2D systems of conservation laws (in particular the shallow water equations) without source terms. The LTS scheme is demonstrated to preserve or even gain accuracy and save computational time with respect to the first order scheme.     

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Modification of basis functions in high order discontinuous Galerkin schemes for advection equation

High order discontinuous Galerkin (DG) discretization schemes are considered for an advection boundary-value problem on 2-D unstructured grids with arbitrary geometry of grid cells. A number of test cases are developed to study the sensitivity of a high order DG scheme to local grid distortion. It will be demonstrated how to modify the formulation of a DG discretization for the advection equation. Our approach allows one to maintain the required accuracy on distorted grids while using a fewer number of basis functions for the solution approximation in order to save computational resources.     

A supraconvergent scheme for the Korteweg-de Vries equation

Summary A finite-difference method for the integration of the Korteweg-de Vries equation on irregular grids is analyzed. Under periodic boundary conditions, the method is shown to be supraconvergent in the sense that, though being inconsistent, it is second order convergent. However, such a convergence only takes place on grids with an odd number of points per period. When a grid with an even number of points is used, the inconsistency of the method leads to divergence. Numerical results backing the analysis are presented.     

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

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method.