Physical Sciences and Mathematics Commons^™

On Skew-Symmetric Splitting And Entropy Conservation Schemes For The Euler Equations, Björn Sjögreen, H.C. Yee

United States National Aeronautics and Space Administration: Publications

The Tadmor type of entropy conservation formulation for the Euler equations and various skew-symmetric splittings of the inviscid flux derivatives are discussed. Numerical stability of high order central and Padé type (centered compact) spatial discretization is enhanced through the application of these formulations. Numerical test on a 2-D vortex convection problem indicates that the stability and accuracy of these formulations using the same high order central spatial discretization are similar for vortex travel up to a few periods. For two to three times longer time integrations, their corresponding stability and accuracy behaviors are very different. The goal of this work …

Comparative Study Of High Order Methods For Subsonic Turbulence Simulation With Stochastic Forcing, Alexei G. Kritsuk, H. C. Yee, Björn Sjögreen, Dmitry Kotov

United States National Aeronautics and Space Administration: Publications

A class of spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen [1, 2] is tested on three-dimensional subsonic turbulence simulations with stochastic forcing. The Euler equations are solved using the Strang operator splitting of the homogeneous part of the equations and the stochastic forcing term, with an ODE solver used to integrate the latter. Both Ducros et al. and Kennedy-Gruber skew-symmetric split formulations of the inviscid flux derivatives are considered to minimize the use of numerical dissipation. The nonlinear filter methods are shown to be numerically stable for this application at least up to …

On Entropy Conservation And Kinetic Energy Preservation Methods, H. C. Yee, Björn Sjögreen

United States National Aeronautics and Space Administration: Publications

The Tadmor-type entropy conservative method using the mathematical logarithmic entropy function and two forms of the Sjögreen & Yee entropy conservative methods using the Harten entropy function are examined for their nonlinear stability and accuracy in very long time integration of the Euler equations of compressible gas dynamics. Following the same procedure as Ranocha [6] these entropy conservative methods can be made kinetic energy preserving with minimum added computational effort. The focus of this work is to examine the nonlinear stability and accuracy of these newly introduced high order entropy conserving and kinetic energy preserving methods for very long time …

High-Order Finite-Difference Nonlinear Filter Methods For Subsonic Turbulence Simulation With Stochastic Forcing, Alexei G. Kritsuk, Dmitry Kotov, Bjorn Sjögreen, Helen C. Yee

United States National Aeronautics and Space Administration: Publications

Numerical stability of high-order fi lter schemes developed by Yee & Sjogreen is tested on thr~dimensional turbulence simulations with stochastic forcing and their performance is compared with that ofTVD and \VENO schemes. The bestperforming filter method employs an eighth-order central base scheme with the Kennedy & Gruber skew-symmetric splitting of the inviscid fttLx derivative, a wavelet-based local flow sensor, a nonlinear filter utilizing the dissipative portion of seventh-order \\'ENO scheme, and an explicit third- or fourth-order RungeKutta t ime integration. \Ve show that the filter scheme is more computational]y efficient and provides a wider spectral bandwidth compared to the seventh-order …

Skew-Symmetric Splitting Of High-Order Central Schemes With Nonlinear Filters For Computational Aeroacoustics Turbulence With Shocks, Bjorn Sjögreen, Helen C. Yee, Alan A. Wray

United States National Aeronautics and Space Administration: Publications

A class of high-order nonlinear filter schemes by Yee et al. (J Comput Phys 150:199–238, 1999), Sjögreen and Yee (J Comput Phys 225:910–934, 2007), and Kotov et al. (Commun Comput Phys 19:273–300, 2016; J Comput Phys 307:189–202, 2016) is examined for long-time integrations of computational aeroacoustics (CAA) turbulence applications. This class of schemes was designed for an improved nonlinear stability and accuracy for long-time integration of compressible direct numerical simulation and large eddy simulation computations for both shock-free turbulence and turbulence with shocks. They are based on the skew-symmetric splitting version of the high-order central base scheme in conjunction with …

High Order Entropy Conservative Central Schemes For Wide Ranges Of Compressible Gas Dynamics And Mhd Flows, Björn Sjögreen, H.C. Yee

United States National Aeronautics and Space Administration: Publications

The Sjogreen and Yee [31]high order entropy conservative numerical method for compressible gas dynamics is extended to include discontinuities and also extended to equations of ideal magnetohydrodynamics (MHD). The basic idea is based on Tadmor’s [40]original work for inviscid perfect gas flows. For the MHD four formulations of the MHD are considered: (a) the conservative MHD, (b) the Godunov [14]non-conservative form, (c) the Janhunen [19]– MHD with magnetic field source terms, and (d) a MHD with source terms by Brackbill and Barnes[5]. Three forms of the high order entropy numerical fluxes for the MHD in the finite difference framework are …

Groundwater Withdrawals Under Drought: Reconciling Grace And Land Surface Models In The United States High Plains Aquifer, Wanshu Nie, Benjamin Zaitchik, Matthew Rodell, Sujay V. Kumar, Martha C. Anderson, Christopher Hain

United States National Aeronautics and Space Administration: Publications

Advanced Land Surface Models (LSM) offer a powerful tool for studying hydrological variability. Highly managed systems, however, present a challenge for these models, which typically have simplified or incomplete representations of human water use. Here we examine recent groundwater declines in the US High Plains Aquifer (HPA), a region that is heavily utilized for irrigation and that is also affected by episodic drought. To understand observed decline in groundwater and terrestrial water storage during a recent multiyear drought, we modify the Noah-MP LSM to include a groundwater irrigation scheme. To account for seasonal and interannual variability in active irrigated area, …

Two Decades Old Entropy Stable Method For The Euler Equations Revisited, Björn Sjögreen, H. C. Yee

United States National Aeronautics and Space Administration: Publications

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference closure of Olsson and Oliger, Gerritsen and Olsson, and Yee et al. [2, 7, 25] is revisited. The entropy splitting is a form of skewsymmetric splitting in terms of the physical entropy of the nonlinear Euler flux derivatives. Central differencing applied to the entropy splitting form of the Euler flux derivatives together with SBP difference operators will, hereafter, be referred to as entropy split schemes.

Corrigendum To “On High Order Finite-Difference Metric Discretizations Satisfying Gcl On Moving And Deforming Grids” [J. Comput.Phys. 265 (2014) 211–220], Björn Sjögreen, H.C. Yee, Marcel Vinokur

United States National Aeronautics and Space Administration: Publications

The authors regret that the already published version contains some typographic errors in the formulas for the Runge-Kutta moving metric time stepping method.

Joint Hierarchical Models For Sparsely Sampled High-Dimensional Lidar And Forest Variables, Andrew O. Finley, Sudipto Banerjee, Yuzhen Zhou, Bruce D. Cook, Chad Babcock

United States National Aeronautics and Space Administration: Publications

Recent advancements in remote sensing technology, specifically Light Detection and Ranging (LiDAR) sensors, provide the data needed to quantify forest characteristics at a fine spatial resolution over large geographic domains. From an inferential standpoint, there is interest in prediction and interpolation of the often sparsely sampled and spatially misaligned LiDAR signals and forest variables. We propose a fully process-based Bayesian hierarchical model for above ground biomass (AGB) and LiDAR signals. The processbased framework offers richness in inferential capabilities, e.g., inference on the entire underlying processes instead of estimates only at pre-specified points. Key challenges we obviate include misalignment between the …

Spurious Behavior Of Shock-Capturing Methods By The Fractional Step Approach: Problems Containing Stiff Source Terms And Discontinuities, H.C. Yee, D.V. Kotov, Wei Wang, Chi-Wang Shu

United States National Aeronautics and Space Administration: Publications

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The present study focuses only on solving the reactive system by the fractional step method using the Strang splitting. Studies shows that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the …

Comparative Study Of High-Order Positivity-Preserving Weno Schemes, D. V. Kotov, Helen C. Yee, Bjorn Sjögreen

United States National Aeronautics and Space Administration: Publications

In gas dynamics and magnetohydrodynamics flows, physically, the density ρ and the pressure p should both be positive. In a standard conservative numerical scheme, however, the computed internal energy is obtained by subtracting the kinetic energy from the total energy, resulting in a computed p that may be negative. Examples are problems in which the dominant energy is kinetic. Negative ρ may often emerge in computing blast waves. In such situations the computed eigenvalues of the Jacobian will become imaginary. Consequently, the initial value problem for the linearized system will be ill posed. This explains why failure of preserving positivity …

High Order Finite Difference Methods With Subcell Resolution For Advection Equations With Stiff Source Terms, Wei Wang, Chi-Wang Shu, H.C. Yee, Björn Sjögreen

United States National Aeronautics and Space Administration: Publications

A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be …

High-Order Finite Difference Methods With Subcell Resolution For 2d Detonation Waves, W. Wang, C.-W. Shu, Helen Yee, Bjorn Sjögreen

United States National Aeronautics and Space Administration: Publications

In simulating hyperbolic conservation laws in conjunction with an inhomogeneous stiff source term, if the solution is discontinuous, spurious numerical results may be produced due to the different time scales of the transport part and the source term. This numerical issue often arises in combustion and high-speed chemical reacting flows.

Our objective in this study is to extend this method to two-dimensional reactive Euler equations. The first step of the proposed fractional step method is the convection step which solves the homogeneous hyperbolic conservation law in which any high-resolution shock-capturing method can be used. The aim in this step is …

Variable High-Order Overset Grid Methods For Mixed Steady And Unsteady Multiscale Viscous Hypersonic Nonequilibrium Flows, A. Lani, Bjorn Sjögreen, Helen Yee, W, D. Henshaw

United States National Aeronautics and Space Administration: Publications

The objective of the current paper † is a further validation of the overset grid capability for high-speed chemical nonequilibrium flows. The current investigation, which is a follow-up on previous work by (Sj¨ogreen & Yee 2009b), (Lani et al. 2010a), (Wang et al. 2009), (Wang et al. 2010), (Lani et al. 2010b), extends the use of variable order methods for both inviscid and viscous chemical nonequilibrium flows with strong shocks on 2D and 3D multiblock overlapping grids. A 5-species and one-temperature air model in chemical nonequilibrium is considered in all cases. Second-order TVD, fifth- or higher-order WENO schemes are applied …

High-Order Finite Difference Methods With Subcell Resolution For Hyperbolic Conservation Laws With Stiff Reaction Terms: Preliminary Results, W. Wang, C.-W. Shu, Helen Yee, Bjorn Sjögreen

United States National Aeronautics and Space Administration: Publications

The motivation for this research stems from the high-speed chemical reacting flows which have stiff reaction terms, where the chemical time scales are often much smaller than the fluid dynamics time scales. It is usually too expensive to resolve all the spatial/ temporal scales if we are only interested in the main flow. On the other hand, insufficient spatial/temporal resolution will cause the speed of propagation of discontinuities to be incorrectly predicted for many numerical methods. This numerical phenomenon was first observed by Colella et al. (1986). Then LeVeque & Yee (1990) showed that a similar spurious propagation phenomenon can …

High-Order Simulation Of Hypersonic Nonequilibrium Flows On Overset Grids, A. Lani, Bjorn Sjögreen, Helen Yee

United States National Aeronautics and Space Administration: Publications

The time-accurate unsteady 3D compressible flow solver ADPDIS3D is supported by a grant from the Department of Energy (DOE) SciDAC program through the Science Application Partnership (SAP) initiative. The objective of this grant is to develop, implement and validate this variable high-order 3-D multiblock overlapping (overset) grid solver for turbulence with strong shocks and density variations. ADPDIS3D includes capabilities for both direct numerical simulation (DNS), resolving all scales of the flow fields, and large eddy simulation (LES) modeling the small turbulent scales. One of the unique features of the code is the ability to perform DNS and LES computations in …

High Order Well-Balanced Schemes And Applications To Non-Equilibrium Flow With Stiff Source Terms, Wei Wang, Chi-Wang Shu, H. C. Yee, Björn Sjögreen

United States National Aeronautics and Space Administration: Publications

The stiffness of the source terms in modeling non-equilibrium ow problems containing finite-rate chemistry or combustion poses additional numerical difficulties beyond that for solving non-reacting flows. A well-balanced scheme, which can preserve certain non-trivial steady state solutions exactly, may help minimize some of these difficulties. In this paper, a simple one dimensional non-equilibrium model with one temperature is considered. We first describe a general strategy to design high order well-balanced finite difference schemes and then study the well-balanced properties of the high order finite difference weighted essentially non-oscillatory (WENO) scheme, modified balanced WENO schemes and various TVD schemes. The advantages …

On Well-Balanced Schemes For Non-Equilibrium Flow With Stiff Source Terms, W. Wang, C.-W. Shu, Helen Yee

United States National Aeronautics and Space Administration: Publications

In the modeling of unsteady reactive problems, the interaction of turbulence with finiterate chemistry introduces a wide range of space and time scales, leading to additional numerical difficulties. A main difficulty stems from the fact that most numerical algorithms used in reacting flows were originally designed to solve non-reacting fluids. As a result, spatial stiffness due to reacting source terms and turbulence/chemistry interaction are major stumbling blocks to numerical algorithm development. One of the important numerical issues is the proper numerical treatment of a system of highly coupled stiff non-linear source terms, which will result in possible spurious steady state …

Nonlinear Filtering In Compact High-Order Schemes, H. C. Yee, Björn Sjögreen

United States National Aeronautics and Space Administration: Publications

The adaptive nonlinear filtering approach for shock/turbulence gas dynamics and magnetohydrodynamics (MHD) flows adopted in our previous work is employed in conjunction with compact high-order methods as the spatial base scheme. The objective is to compare the performance of nonlinear filtering in compact high-order schemes with nonlinear filtering in standard central (non-compact) schemes for multiscale problems containing shock waves.

Entropy Splitting For High Order Numerical Simulation Of Vortex Sound At Low Mach Numbers, Bernhard Müller, H. C. Yee

United States National Aeronautics and Space Administration: Publications

Several recent developments in efficient, stable, highly parallelizable high order non-dissipative spatial schemes with characteristic based filters that exhibit low dissipation for long time linear and nonlinear wave propagations are utilized for computational aeroacoustics (CAA). For stability consideration, the Euler equations are split into a conservative and a symmetric non-conservative portion. Due to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the split Euler equations are formulated in perturbation form to minimize numerical cancellation errors. Spurious oscillations are suppressed by a characteristic-based filter. The method has been applied to accurately simulate the sound emitted by …

Building Blocks For Reliable Complex Nonlinear Numerical Simulations, Helen Yee

United States National Aeronautics and Space Administration: Publications

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is …

Designing Adaptive Low-Dissipative High Order Schemes For Long-Time Integrations, Helen Yee, Bjorn Sjögreen

United States National Aeronautics and Space Administration: Publications

A general framework for the design of adaptive low-dissipative high order schemes is presented. It encompasses a rather complete treatment of the numerical approach based on four integrated design criteria: (1) For stability considerations, condition the governing equations before the application of the appropriate numerical scheme whenever it is possible. (2) For consistency, compatible schemes that possess stability properties, including physical and numerical boundary condition treatments, similar to those of the discrete analogue of the continuum are preferred. (3) For the minimization of numerical dissipation contamination, efficient and adaptive numerical dissipation control to further improve nonlinear stability and accuracy should …

Entropy Splitting For High Order Numerical Simulation Of Vortex Sound At Low Mach Numbers, B. Müller, H.C. Yee

United States National Aeronautics and Space Administration: Publications

A method of minimizing numerical errors, and improving nonlinear stability and accuracy associated with low Mach number computational aeroacoustics (CAA) is proposed. The method consists of two levels.

Semi-Implicit And Fully Implicit Shock-Capturing Methods For Nonequilibrium Flows, Helen C. Yee, Judy L. Shinn

United States National Aeronautics and Space Administration: Publications

Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogeneous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a more efficient solution procedure. The primary motivation for constructing these schemes was to address large systems of thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be …

A Class Of High-Resolution Explicit And Implicit Shock-Capturing Methods, Helen C. Yee

United States National Aeronautics and Space Administration: Publications

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shockcapturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient …

Upwind And Symmetric Shock- Capturing Schemes, H.C. Yee

United States National Aeronautics and Space Administration: Publications

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis will be on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the exact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently …

Implicit Tvd Schemes For Hyperbolic Conservation Laws In Curvilinear Coordinates, Helen Yee, A. Harten

United States National Aeronautics and Space Administration: Publications

A one-parameter family of explicit and implicit upwind second-order-accurate, total variation diminishing (TVD) schemes has been developed by Harten. These TVD schemes have the property of not generating spurious oscillations when applied to one-dimensional nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. The goal of this work is to extend these methods to the multidimensional hyperbolic conservation laws in curvilinear coordinates. Various ways of linearizing the implicit operator and solution strategies to improve the computation efficiency of the implicit algorithm are discussed. Numerical experiments with some AGARD test cases for steady-state airfoil calculations show that the proposed linearized …

Implicit Total Variation Diminishing (Tvd) Schemes For Steady-State Calculations, H. C. Yee, R. F Warming, A. Harten

United States National Aeronautics and Space Administration: Publications

We examine the application of a new implicit unconditionally-stable high-resolution TVD scheme to steady-state calculations. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by Harten for the computation of weak solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a fairly rapid convergence rate, but also generates a highly-resolved approximation to the steady-state solution. A detailed implementation of the implicit scheme for the one- and two-dimensional compressible …

Boundary Approximations For Implicit Schemes For .One-Dimensional Inviscid Equations Of Gasdynamics, Helen Yee, R. M. Beam, R. F. Warming

United States National Aeronautics and Space Administration: Publications

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the numerical experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit boundary schemes, and unconditionally stable schemes.