Efficient upwind algorithms for solution of the Euler and Navier-stokes equations

An efficient three-dimensionasl tructured solver for the Euler and Navier-Stokese quations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed

Object-oriented hyperbolic solver on 2D-unstructured meshes applied to the shallow water equations

Fluid dynamics, like other physical sciences, is divided into theoretical and experimental branches. However, computational fluid dynamics (CFD) is third branch of Fluid dynamics, which has aspects of both the previous two branches. CFD is a supplement rather than a replacement to the experiment or theory. It turns a computer into a virtual laboratory, providing insight, foresight, return on investment and cost savings1. This work is a step toward an approach that realise a new and effective way of developing these CFD models

쌍곡 보존 법칙들을 풀기 위한 고차정확도 수치기법에 대한 연구

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 강명주.In this thesis, we develop efficient and high order accurate numerical schemes for solving hyperbolic conservation laws such as the Euler equation and the ideal MHD(Magnetohydrodynamics) equations. The first scheme we propose is the \textit{wavelet-based adaptive WENO method}. The Finite difference WENO scheme is one of the popular numerical schemes for application to hyperbolic conservation laws. The scheme has high order accuracy, robustness and stable property. On the other hand, the WENO scheme is computationally expensive since it performs characteristic decomposition and computes non-linear weights for WENO interpolations. In order to overcome the drawback, we propose the adaptation technique that applies WENO differentiation for only discontinuous regions and central differentiation without characteristic decomposition for the other regions. Therefore continuous and discontinuous regions should be appropriately classified so that the adaptation method successfully works. In the wavelet-based WENO method, singularities are detected by analyzing wavelet coefficients. Such coefficients are also used to reconstruct the compressed informations. Secondly, we propose \textit{central-upwind schemes with modified MLP(multi-dimensional limiting process)}. This scheme decreases computational cost by simplifying the scheme itself, while the first method achieve efficiency by skipping grid points. Generally the high-order central difference schemes for conservation laws have no Riemann solvers and characteristic decompositions but tend to smear linear discontinuities. To overcome the drawback of central-upwind schemes, we use the multi-dimensional limiting process which utilizes multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications.1 Introduction 1 2 Governing Equations 7 2.1 Hyperbolic Conservation Laws 7 2.2 Euler equation 9 2.2.1 Model equation 9 2.2.2 Eigen-structure 10 2.3 Ideal MHD equation 14 2.3.1 Model equation 14 2.3.2 Eigen-Structure 15 2.4 The r B = 0 Constraint in MHD Codes 20 2.4.1 Constraints Transport Method 20 2.4.2 Divergence cleaning technique 23 3 Wavelet-based Adaptation Strategy with Finite Dierence WENO scheme 28 3.1 Finite Dierence WENO scheme 28 3.1.1 Characteristic Decomposition 28 3.1.2 WENO-type Approximations 30 3.2 Wavelet Analysis 32 3.2.1 Multi-resolution Approximations 32 3.2.2 Orthogonal Wavelets 36 3.2.3 Constructing Wavelets 37 3.2.4 Biorthogonal Wavelets 38 3.2.5 Interpolating Scaling Function 40 3.3 Adaptive wavelet Collocation Method 45 3.3.1 Interpolating Wavelets 47 3.3.2 Lifting Scheme 52 3.3.3 Lifting Donoho wavelets family 56 3.3.4 The Lifted interpolating wavelet transform 58 3.3.5 Compression 64 3.4 Wavelet-based Adaptive WENO scheme 65 3.4.1 Adjacent Zone 65 3.4.2 Methodology for Spatial discretizations 66 3.4.3 Time Integration 67 3.4.4 Conservation error and boundary treatment 68 3.4.5 Overall Process 69 3.5 Numerical results 69 3.5.1 1-dimensional equations 70 3.5.2 2-dimensional Euler equations 71 3.5.3 2-dimensional MHD equations 83 4 Combination of Central-Upwind Method and Multi-dimensional Limiting Process 90 4.1 Review of Central-Upwind method 92 4.2 Review of Multi-dimensional Limiting Process 95 4.3 Central-Upwind method with Modied MLP limiter 98 4.4 Numerical results 104 4.4.1 Linear advection equation 105 4.4.2 Burger's equation 106 4.4.3 2D Euler system - Four shocks 106 4.4.4 2D Euler system - Rayleigh-Taylor instability 107 4.4.5 2D Euler system - Double Mach reection of a strong shock 109 5 Conclusions 111 Abstract (in Korean) 121Docto

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

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes