Fifth-Order Mapped Semi-Lagrangian Weighted Essentially Nonoscillatory Methods Near Certain Smooth Extrema

Fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semi-Lagrangian finite volume (M-SL-FV) WENO 5 method and the mapped compact semi-Lagrangian finite difference (M-C-SL-FD) WENO 5 method. The weights in the more common scheme lose accuracy at certain smooth extrema. We introduce mapped weighting to handle the problem. In general, a cell average is applied to construct the M-SL-FV WENO 5 reconstruction, and the M-C-SL-FD WENO 5 interpolation scheme is proposed based on an interpolation approach. An accuracy test and numerical examples are used to demonstrate that the two schemes reduce the loss of accuracy and improve the ability to capture discontinuities

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-resolution numerical schemes for compressible flows and\ud compressible two-phase flows

Several high-resolution numerical schemes based on the Constrained Interpolation Profile Conservative Semi-Lagrangian (CIP-CSL), Essentially Non-Oscillatory (ENO), Weighted ENO (WENO), Boundary Variation Diminishing (BVD), and Tangent of Hyperbola for INterface Capturing (THINC) schemes have been proposed for compressible flows and compressible two-phase flows. In the first part of the thesis, three high-resolution CIP-CSL schemes are proposed. (i) A fully conservative and less oscillatory multi-moment scheme (CIP-CSL3-ENO) is proposed based on two CIP-CSL3 schemes and the ENO scheme. An ENO indicator is designed to intentionally select non-smooth stencil but can efficiently minimise numerical oscillations. (ii) Motivated by the observation that combining two different types of reconstruction functions can effectively reduce numerical diffusion and oscillations, a better-suited scheme CIP-CSL-ENO5 is proposed based on hybrid-type CIP-CSL reconstruction functions and a newly designed ENO indicator. (iii) To further reduce the numerical diffusion in vicinity of discontinuities, the BVD and THINC schemes are implemented in the CIP-CSL framework. The resulting scheme accurately capture both smooth and discontinuous solutions simultaneously by selecting an appropriate reconstruction function. In the second part of the thesis, the TWENO (Target WENO) scheme is proposed to improve the accuracy of the fifth-order WENO scheme. Unlike conventional WENO schemes, the TWENO scheme is designed to restore the highest possible order interAbstract iv polation when three sub-stencils or two adjacent sub-stencils are smooth. To further minimise the numerical diffusion across discontinuities, the TWENO scheme is implemented with the THINC scheme and the Total Boundary Variation Diminishing (TBVD) algorithm. The resulting scheme TBVD-TWENO-THINC is also applied to solve the five-equation model for compressible two-phase flows. Verified through a wide range of benchmark tests, the proposed numerical schemes are able to obtain accurate and high-resolution numerical solutions for compressible flows and compressible two-phase flows

ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA)

The proceedings of the Benchmark Problems in Computational Aeroacoustics Workshop held at NASA Langley Research Center are the subject of this report. The purpose of the Workshop was to assess the utility of a number of numerical schemes in the context of the unusual requirements of aeroacoustical calculations. The schemes were assessed from the viewpoint of dispersion and dissipation -- issues important to long time integration and long distance propagation in aeroacoustics. Also investigated were the effect of implementation of different boundary conditions. The Workshop included a forum in which practical engineering problems related to computational aeroacoustics were discussed. This discussion took the form of a dialogue between an industrial panel and the workshop participants and was an effort to suggest the direction of evolution of this field in the context of current engineering needs

Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games

In this dissertation we consider two major subjects. The primary topic is the Entropy Viscosity method for Lagrangian hydrodynamics, the goal of which is to solve numerically the Euler equations of compressible gas dynamics. The second topic is concerned with applications of second order central differencing schemes to the Mean Field Games equations. The Entropy Viscosity method discretizes all kinematic and thermodynamic variables by high-order finite elements and solves the resulting discrete problem on a computational mesh that moves with the material velocity. The method is based on two major concepts. The first one is producing high order convergence rates for smooth solutions even with active viscosity terms. This is achieved by using high order finite element spaces and, more importantly, entropy based viscosity coefficients that clearly distinguish between smooth and singular regions. The second concept is providing control over oscillations around contact discontinuities as well as oscillations in shock regions. Achieving this requires adding extra viscosity terms in a way that the resulting system is still in agreement with generalized entropy inequalities, the minimum principle on the specific entropy and the general requirements for artificial tensor viscosities like orthogonal transformation invariance, radial symmetry, Galilean invariance, etc. We define a fully-discrete finite element algorithm and present numerical results on model Lagrangian hydro problems. We also discuss possible extensions of the method, e.g. length scale independent viscosity coefficients, incorporating mass diffusion into the mesh motion, and handling of different materials. In addition we present approaches to the different stages of arbitrary Lagrangian-Eulerian (ALE) methods, which can be used to extend the Entropy Viscosity method. That is, we discuss mesh relaxation by harmonic smoothing schemes, advection based solution remap, and multi-material zones treatment. The Mean Field Games (MFG) equations describe situations in which a large number of individual players choose their optimal strategy by considering global (but limited) incentive information that is available to everyone. The resulting system consists of a forward Hamilton-Jacobi equation and a backward convection-diffusion equation. We propose fully discrete explicit second order staggered finite difference schemes for the two equations and combine these schemes into a fixed point iteration algorithm. We discuss the second order accuracy of both schemes, their interaction in time, memory issues resulting from the forward-backward coupling, stopping criteria for the fixed point iteration, and parallel performance of the method

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Neutron stars in numerical relativity

Binary neutron star coalescence lead to extremely violent merger events in the universe. Such events give rise to a variety of observable phenomena in both the gravitational and the electromagnetic channels. In this thesis we study such extreme events in the last milliseconds around the merger. Gravity and its coupling to other physics for such systems is described by Einsteins theory of general relativity. Since these are highly nonlinear coupled set of partial differential equations, no analytical solutions exist for generic and dynamical systems such as binary neutron stars in the strong-field regime. Therefore, the usage of numerical methods is inevitable. In this thesis we consider configurations of binary neutron stars with varying equation of state, spin, eccentricity and spin orientation. In particular we present the first numerical relativity simulations of highly eccentric binary neutron stars in full (3+1)D using consistent initial data, i.e., setups which are in agreement with the Einstein equations and with the equations of general relativistic hydrodynamics in equilibrium. Additionally, we also study precessing binary neutron star systems using full (3+1)D numerical relativity simulations with consistent initial data. Further, we investigate a new numerical approach for solving the equations of general relativistic hydrodynamics using the discontinuous Galerkin method. It combines the key aspects of finite volume and traditional finite element methods, for e.g., element locality and hp adaptivity besides the benefit of spectral convergence for smooth solutions. Discontinuous Galerkin methods ease the treatment of complex grid geometries and, allow simple and efficient parallelization