HIGH ORDER SHOCK CAPTURING SCHEMES FOR HYPERBOLIC CONSERVATION LAWS AND THE APPLICATION IN OPEN CHANNEL FLOWS

Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows

Higher Order A-Stable Schemes for the Wave Equation Using a Successive Convolution Approach

In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL$^T$ formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient $O(N)$ convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers which are unconditionally stable, accurate of order $2P$, and require $O(P^d N)$ operations per time step, where $N$ is the number of spatial points and $d$ the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs and can also be extended to parabolic PDEs

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

AbstractThe 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

Alternating direction implicit time integrations for finite difference acoustic wave propagation: Parallelization and convergence

This work studies the parallelization and empirical convergence of two finite difference acoustic wave propagation methods on 2-D rectangular grids, that use the same alternating direction implicit (ADI) time integration. This ADI integration is based on a second-order implicit Crank-Nicolson temporal discretization that is factored out by a Peaceman-Rachford decomposition of the time and space equation terms. In space, these methods highly diverge and apply different fourth-order accurate differentiation techniques. The first method uses compact finite differences (CFD) on nodal meshes that requires solving tridiagonal linear systems along each grid line, while the second one employs staggered-grid mimetic finite differences (MFD). For each method, we implement three parallel versions: (i) a multithreaded code in Octave, (ii) a C++ code that exploits OpenMP loop parallelization, and (iii) a CUDA kernel for a NVIDIA GTX 960 Maxwell card. In these implementations, the main source of parallelism is the simultaneous ADI updating of each wave field matrix, either column-wise or row-wise, according to the differentiation direction. In our numerical applications, the highest performances are displayed by the CFD and MFD CUDA codes that achieve speedups of 7.21x and 15.81x, respectively, relative to their C++ sequential counterparts with optimal compilation flags. Our test cases also allow to assess the numerical convergence and accuracy of both methods. In a problem with exact harmonic solution, both methods exhibit convergence rates close to 4 and the MDF accuracy is practically higher. Alternatively, both convergences decay to second order on smooth problems with severe gradients at boundaries, and the MDF rates degrade in highly-resolved grids leading to larger inaccuracies. This transition of empirical convergences agrees with the nominal truncation errors in space and time.Comment: 20 pages, 5 figure

ULTRA-SHARP nonoscillatory convection schemes for high-speed steady multidimensional flow

For convection-dominated flows, classical second-order methods are notoriously oscillatory and often unstable. For this reason, many computational fluid dynamicists have adopted various forms of (inherently stable) first-order upwinding over the past few decades. Although it is now well known that first-order convection schemes suffer from serious inaccuracies attributable to artificial viscosity or numerical diffusion under high convection conditions, these methods continue to enjoy widespread popularity for numerical heat transfer calculations, apparently due to a perceived lack of viable high accuracy alternatives. But alternatives are available. For example, nonoscillatory methods used in gasdynamics, including currently popular TVD schemes, can be easily adapted to multidimensional incompressible flow and convective transport. This, in itself, would be a major advance for numerical convective heat transfer, for example. But, as is shown, second-order TVD schemes form only a small, overly restrictive, subclass of a much more universal, and extremely simple, nonoscillatory flux-limiting strategy which can be applied to convection schemes of arbitrarily high order accuracy, while requiring only a simple tridiagonal ADI line-solver, as used in the majority of general purpose iterative codes for incompressible flow and numerical heat transfer. The new universal limiter and associated solution procedures form the so-called ULTRA-SHARP alternative for high resolution nonoscillatory multidimensional steady state high speed convective modelling

Three-dimensional unsteady viscous flow analysis over airfoil sections

A three-dimensional solution procedure for the approximate form of the Navier-Stokes equation was exercised in the two- and three-dimensional modes to compute the unsteady turbulent boundary layer on a flat plate corresponding to the data of Karlsson. The procedure is based on the use of a consistently split Linearized Block Implicit technique in conjunction with a QR operator scheme. New time-dependent upstream boundary conditions were developed that yielded realistic solutions for the interior in the vicinity of the upstream boundary. Comparisons of the computation employing these boundary conditions with the data indicate that both qualitative and quantitative agreement was obtained for the mean velocity and the in phase and out of phase components of the first harmonic of the velocity. In addition, the calculation gave results for the skin friction phase angle that had expected physical behavior for large distances downstream of the inflow boundary. For the three-dimensional case, the two-dimensional data of Karlsson was considered, but in a coordinate system skewed at 45 deg to the free stream direction. The results of the calculations were in excellent agreement with the data and the two-dimensional computations