Part 1: Numerical Modeling for Compressible Plasma Flows
|
|
- Noreen Eaton
- 5 years ago
- Views:
Transcription
1 Part 1: Numerical Modeling for Compressible Plasma Flows Dongwook Lee Applied Mathematics & Statistics University of California, Santa Cruz AMS 280C Seminar October 17, 2014 MIRA, BG/Q, Argonne National Lab 49,152 nodes, 786,432 cores FLASH Simulation of a 3D Core-collapse Supernova Courtesy of S. Couch
2 Scientific Goal To develop solution accurate, efficient, and stable numerical algorithms for a wide range of astrophysical regimes using high-performance computer simulations Type Ia SN Mira, BG/Q at ALCF Core Collapse SN
3 First Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
4 Preliminaries Astrophysical flows are highly compressible for many purposes, involving shocks and discontinuities. It is fair to say that 99% of the readily observed universe is in plasma state. The interstellar medium, stars, and exotic compact objects are all composed of or surrounded by ionized matter. Knowledge of compressible plasmas and their relevant numerical treatments are essential to understand the universe. The gained knowledge extends to understand high-energy-density physics (a.k.a., laboratory astrophysics).
5 Plasma A plasma is a macroscopically electrically neutral substance containing many interacting free electrons and ions which exhibit collective behavior due to the long-range Coulomb forces. The interactions of a magnetic field with a plasma play crucial roles in plasma physics.
6 Plasma Our Sun emits a highly conducting tenuous plasma, called solar wind, a consequence of the hot corona (1~2 million K). Solar wind Ongoing researches for safe controlled thermonuclear fusion energies such as tokomak, z-pinch, laser inertial confinement fusion, are all based on plasma physics. tokomak
7 Two Ways to Model Plasma Kinetic Theory (microscopic): a. adopts a kinetic description with distribution functions f to represent particles, governed by the Boltzmann equations in the phase space and time: b. fully kinetic, PIC; gyrokinetic Fluid Description (macroscopic): a. multi-fluid theory (e.g., two-fluid considers electrons and ions) b. single-fluid theory (MHD treats plasma as a whole)
8 MHD Flow Regimes Given the generalized Ohm s law: For typical large-scale, low-frequency plasma condition, one can approximate: (1) ideal MHD: (2) resistive MHD: (3) Hall MHD: (4) BBT MHD: (see the next talk!)
9 Divergence-free B-fields One very important property in MHD is to satisfy: r B =0 Consider the Lorentz force per unit volume: If the solenoidal constraint holds, then the Lorentz force becomes conservative: Otherwise, the Lorentz force is not conservative and MHD equations violate conservation laws!
10 Compressible Solvers Godunov-type formulation, based on solving Riemann problems (RP), is a very good numerical method to model compressible flows. Godunov-type techniques are based on the finite volume (FV) scheme, which describes PDEs in integral forms. The integral form of FV discretization allows weak solutions (i.e., discontinuous solutions such as shocks and discontinuities), and hence automatically satisfies the conservation property. Note, in general, the Lax equivalence theorem (LET: convergence iff stability + consistency) is only valid for smooth solutions, meaning that, not all numerical schemes (e.g., FD, FE) could correctly approximate weak solutions. But LET holds for FV.
11 Second Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
12 Scientific Tasks Science Problem (IC, BC, ODE/PDE) Simulator (code, computer) Results (Validation, verification, analysis)
13 1. Mathematical Models Hydrodynamics + r ( v) =0 mass v + r ( vv)+rp = g momentum eqn + r [( E + P )v] = v g total + r [( + P )v] v rp =0 Equation of State P =( 1) E = v 2
14 2. Mathematical Models + r ( v) =0 mass eqn + r ( vv BB)+rP = g + r momentum eqn + r [v( E + P ) B(v B)] = g v + r (v + rt )+r (B ( r B)) total + r (vb Bv) = r ( r B) induction eqn E = v B2 2 P = p + B2 2 Equation of State = µ[(rv)+(rv) T 2 (r v)i] 3 viscosity solenodidal constraint r B =0
15 Divergence-free B-fields Different numerical MHD schemes depending on how you control r B =0condition. This is a big research field in MHD. Ignore! Erroneous plasma transport orthogonal to B-field (Brackbill and Barnes, 1980); Eigenvector degeneracy (Crockett et al., 2005) 8-wave (Powell et al., 1999) Projection (Brackbill and Barnes, 1980; Ryu et al., 1995; Balsara 1998; Crockett et al., 2005) Hyperbolic/parabolic cleaning (Dedner et al., 2002) Constrained-transport (Evans and Hawely, 1988; Balsara and Spicer, 1999; Gardiner and Stone, 2005; Lee and Deane, 2009)
16 3. Mathematical Models HEDP: Separate energy eqns for ion, electron, radiation ( 3-temperature, or ( ion)+r ( ion v)+p ion r v = c v,ele ei (T ele T ion ) ( ele)+r ( ele v)+p ele r v = c v,ele ei (T ion T ele ) r q ele + Q abs Q emis + Q las ( rad)+r ( rad v)+p rad r v = r q rad Q abs + Q emis radiation energy tot = ion + ele + rad P tot,t ion,t ele,t rad = EoS(, ion, ele, rad ) 3T EoS Compare 3T with a simple ( tot)+r ( tot v)+p tot r v =0 P tot = EoS(, tot ) T ion = T ele = T rad, or T ele = T ion,t rad =0
17 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
18 Finite Volume Formulations Integral form of PDE: Z xi+1/2 x i 1/2 u(x, t n+1 )dx Z xi+1/2 x i 1/2 u(x, t n )dx = Z tn+1 t n f(u(x i 1/2,t))dt Z tn+1 t n f(u(x i+1/2,t))dt Volume averaged, cell-centered quantity & time averaged flux: U n i = 1 x Z xi+1/2 and x i 1/2 u(x, t n )dx F n i 1/2 = 1 t Finite wave speed in hyperbolic system: Z tn+1 t n f(u(x i 1/2,t))dt F n i 1/2 = F(U n i 1,U n i ) * High-order reconstruction in space & time * Riemann problem at each cell-interface, i-1/2 General discrete difference equation in conservation form in 1D: U n+1 i = U n i t x (F n i+1/2 F n i 1/2 )
19 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
20 High-Order Polynomial Reconstruction FOG PLM PPM Godunov s order-barrier theorem (1959) Monotonicity-preserving advection schemes are at most first-order! (Oh no ) Only true for linear PDE theory (YES!) High-order polynomial schemes became available using non-linear slope limiters (70 s and 80 s: Boris, van Leer, Zalesak, Colella, Harten, Shu, Engquist, etc) Can t avoid oscillations completely (non-tvd) Instability grows (numerical INSTABILITY!)
21 Low vs. High order Reconstructions
22 Traditional High-Order Schemes Traditional approaches to get Nth high-order schemes take (N-1)th degree polynomial for interpolation/reconstruction only for normal direction (e.g., PLM, PPM, ENO, WENO, etc) with monotonicity controls (e.g., slope limiters, artificial viscosity) High-order in FV is tricky (when compared to FD) volume-averaged quantities (quadrature rules) preserving conservation w/o losing accuracy higher the order, larger the stencil high-order temporal update (ODE solvers, e.g., RK3, RK4, etc.) 2D stencil for 2nd order PLM 2D stencil for 3rd order PPM
23 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
24 Riemann Problem & Godunov Method The Riemann problem: Two cases: PDEs: U t + AU x =0, 1 <x<1,t>0 ( IC : U(x, t = 0) = U 0 U L if x<0, (x) = if x>0. U R Shock solution Rarefaction solution
25 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
26 A Discrete World of FV U(x, t n ) x i 1 x i x i+1
27 A Discrete World of FV piecewise polynomial reconstruction on each cell u(x i,t n )=P i (x),x2 (x i 1/2,x i 1/2 ) x i 1 x i x i+1 u R = P i (x i+1/2 ) u L = P i+1 (x i+1/2 )
28 A Discrete World of FV At each interface we solve a RP and obtain F i+1/2 x i 1 x i x i+1
29 A Discrete World of FV We are ready to advance our solution in time and get new volume-averaged states U n+1 i = U n i t x (F i+1/2 F i 1/2 )
30 Various Reconstructions Low-order 1st scheme 1st on 400 cells 2nd 3rd High-order scheme 5th 1st on 800 cells 200 cells
31 Various Reconstructions
32 Various Reconstructions PLM PPM WENO-5 WENO-Z PLM+Roe
33 Various Riemann Solvers HLLC: 3rd most diffusive Roe: least diffusive LLF: most diffusive HLL: 2nd most diffusive
34 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
35 Multidimensional Formulation 2D discrete difference equation in conservation form: U n+1 i,j = U n i,j t x (F n i+1/2,j F n i 1/2,j ) t y (Gn i,j+1/2 G n i,j 1/2 ) Two different approaches: directionally split formulation update each spatial direction separately, easy to implement, robust always good? directionally unsplit formulation update both spatial directions at the same time, harder to implement you gain extra from what you pay for
36 Unsplit FV Formulation 2D discrete difference equation in conservation form: U n+1 i,j = U n i,j (a) 1st order donor cell t x (F i+1/2,j n Fi n 1/2,j ) t y (Gn i,j+1/2 G n i,j 1/2 ) (b) 2nd order corner-transport-upwind (CTU) (i, j) (i, j) U n+1 i,j = U n i,j u t x [U n i,j U n i 1,j] v t y [U i,j n Ui,j n 1] U n+1 i,j + t2 2 = U n i,j n u v x y (U i,j n Ui,j n 1) + v v y y (U i,j n Ui n 1,j) u t x [U i,j n Ui n v t 1,j] y [U i,j n Ui,j n 1] vy (U i n 1,j Ui n 1,j 1) v y (U n i,j 1 U n i 1,j 1) o Extra cost for corner coupling!
37 Unsplit FV Formulation 2D discrete difference equation in conservation form: U n+1 i,j = U n i,j (a) 1st order donor cell t x (F i+1/2,j n Fi n 1/2,j ) t y (Gn i,j+1/2 G n i,j 1/2 ) (b) 2nd order corner-transport-upwind (CTU) (i, j) (i, j) u t x + v t u t y apple 1 max x, v t y Smaller stability region apple 1 Gain: Extended stability region
38 Unsplit vs. Split Split PPM Single-mode RT instability (Almgren et al. ApJ, 2010) Split solver: High-wavenumber instabilities grow due to experiencing high compression and expansion in each directional sweep Unsplit PPM Unsplit solver: High-wavenumber instabilities are suppressed and do not grow For MHD, it is more crucial to use unsplit in order to preserve divergence-free solenoidal constraint (Lee & Deane, 2009; Lee, 2013): r B =0
39 Unsplit vs. Split unsplit PPM split PPM
40 Unsplit vs. Split: u@b =0 w( B x x + B y y )=wr B Situation is more critical in MHD (Gardiner & Stone, 2005; Lee & Deane, 2009) Split solver: Simply fails to preserve the solenoidal constraint of magnetic fields because one cannot balance the cancellation from separate sweeps of x and y. The error will increase Bz in time if w is not zero. Unsplit solver: Dynamics of in-plane magnetic fields satisfy the divergence-free constraint IF correctly implemented (Lee & Deane, 2009; Lee 2013)
41 Third Episode 1. Plasma & MHD 2. Governing Equations 3. Finite Volume MHD (a) Reconstruction (b) Riemann Problem (c) Unsplit vs. Split (d) divb=0
42 Constrained Transport MHD Solves induction equations on staggered grid (duality relation should be mentioned with the picture) early time Div B = 0 to machine accuracy Finite volume Godunov algorithms gives electric fields at face centers later time 1. arithmetic averaging (Balsara & Spicer, 1999) 2. plane-parallel, grid-aligned reconstruction (Gardiner & Stone, 2005) Bad oscillations! 3. high-order interpolation (Lee & Deane, 2009) B n+1 x,i+1/2,j = Bn x,i+1/2,j t n E n+1/2 y z,i+1/2,j+1/2 o E n+1/2 z,i+1/2,j 1/2, B n+1 y,i,j+1/2 = Bn y,i,j+1/2 t n x E n+1/2 z,i+1/2,j+1/2 + En+1/2 z,i 1/2,j+1/2 o.
43 1. Arithmetic averaging CT Consider u>0; v 0 Weakly magnetized field loop advection test Gardiner & Stone (2005); Lee & Deane (2009); Lee (2013) for small angle advection Balsara & Spicer, 1999, JCP
44 2. Contact-Mode-Upwind CT Lack of numerical dissipations generate unphysical instabilities! Gardiner & Stone, JCP, 2005
45 Consider u>0; v 0 3. Improved Upwind CT only upwind! Lee, JCP, 2013
46 3. Improved Upwind CT Important to advect magnetized flow in a stable manner Upwind-biased scheme improves numerical stability in FL advection (Lee, JCP, 2013)
47 Summary Proper numerical schemes need to be carefully chosen for different set of physics for accuracy, efficiency and stability. Do not blindly believe numerics unless you know what you do with them. Different numerical approaches can give very different results on a given problem.
The RAMSES code and related techniques 2- MHD solvers
The RAMSES code and related techniques 2- MHD solvers Outline - The ideal MHD equations - Godunov method for 1D MHD equations - Ideal MHD in multiple dimensions - Cell-centered variables: divergence B
More informationThe Center for Astrophysical Thermonuclear Flashes. FLASH Hydrodynamics
The Center for Astrophysical Thermonuclear Flashes FLASH Hydrodynamics Jonathan Dursi (CITA), Alan Calder (FLASH) B. Fryxell, T. Linde, A. Mignone, G. Wiers Many others! Mar 23, 2005 An Advanced Simulation
More informationA Comparative Study of Divergence-Cleaning Techniques for Multi-Dimensional MHD Schemes )
A Comparative Study of Divergence-Cleaning Techniques for Multi-Dimensional MHD Schemes ) Takahiro MIYOSHI and Kanya KUSANO 1) Hiroshima University, Higashi-Hiroshima 739-856, Japan 1) Nagoya University,
More informationVarious Hydro Solvers in FLASH3
The Center for Astrophysical Thermonuclear Flashes Various Hydro Solvers in FLASH3 Dongwook Lee FLASH3 Tutorial June 22-23, 2009 An Advanced Simulation and Computing (ASC) Academic Strategic Alliances
More informationComputational Astrophysics
16 th Chris Engelbrecht Summer School, January 2005 3: 1 Computational Astrophysics Lecture 3: Magnetic fields Paul Ricker University of Illinois at Urbana-Champaign National Center for Supercomputing
More informationState of the Art MHD Methods for Astrophysical Applications p.1/32
State of the Art MHD Methods for Astrophysical Applications Scott C. Noble February 25, 2004 CTA, Physics Dept., UIUC State of the Art MHD Methods for Astrophysical Applications p.1/32 Plan of Attack Is
More informationNumerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement
Numerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement Romain Teyssier CEA Saclay Romain Teyssier 1 Outline - Euler equations, MHD, waves, hyperbolic
More informationThe RAMSES code and related techniques I. Hydro solvers
The RAMSES code and related techniques I. Hydro solvers Outline - The Euler equations - Systems of conservation laws - The Riemann problem - The Godunov Method - Riemann solvers - 2D Godunov schemes -
More informationMulti-D MHD and B = 0
CapSel DivB - 01 Multi-D MHD and B = 0 keppens@rijnh.nl multi-d MHD and MHD wave anisotropies dimensionality > 1 non-trivial B = 0 constraint even if satisfied exactly t = 0: can numerically generate B
More informationA Finite Volume Code for 1D Gas Dynamics
A Finite Volume Code for 1D Gas Dynamics Michael Lavell Department of Applied Mathematics and Statistics 1 Introduction A finite volume code is constructed to solve conservative systems, such as Euler
More informationABSTRACT. A new unsplit staggered mesh algorithm (USM) that solves multidimensional
ABSTRACT Title of dissertation: An Unsplit Staggered Mesh Scheme For Multidimensional Magnetohydrodynamics: A Staggered Dissipation-control Differencing Algorithm Dongwook Lee, Doctor of Philosophy, 2006
More informationRiemann Solvers and Numerical Methods for Fluid Dynamics
Eleuterio R Toro Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introduction With 223 Figures Springer Table of Contents Preface V 1. The Equations of Fluid Dynamics 1 1.1 The Euler
More informationPREPRINT November 5, 2007
AN UNSPLIT STAGGERED MESH SCHEME FOR MULTIDIMENSIONAL MAGNETOHYDRODYNAMICS WITH EFFICIENT DISSIPATION CONTROLS Dongwook Lee a and Anil Deane b, a ASC FLASH Center, University of Chicago, 5640 S. Ellis,
More informationExtremum-Preserving Limiters for MUSCL and PPM
arxiv:0903.400v [physics.comp-ph] 7 Mar 009 Extremum-Preserving Limiters for MUSCL and PPM Michael Sekora Program in Applied and Computational Mathematics, Princeton University Princeton, NJ 08540, USA
More informationChapter 1. Introduction
Chapter 1 Introduction Many astrophysical scenarios are modeled using the field equations of fluid dynamics. Fluids are generally challenging systems to describe analytically, as they form a nonlinear
More informationA Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws
A Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws Kilian Cooley 1 Prof. James Baeder 2 1 Department of Mathematics, University of Maryland - College Park 2 Department of Aerospace
More informationGodunov methods in GANDALF
Godunov methods in GANDALF Stefan Heigl David Hubber Judith Ngoumou USM, LMU, München 28th October 2015 Why not just stick with SPH? SPH is perfectly adequate in many scenarios but can fail, or at least
More informationComparison of (Some) Algorithms for Edge Gyrokinetics
Comparison of (Some) Algorithms for Edge Gyrokinetics Greg (G.W.) Hammett & Luc (J. L.) Peterson (PPPL) Gyrokinetic Turbulence Workshop, Wolfgang Pauli Institute, 15-19 Sep. 2008 w3.pppl.gov/~hammett Acknowledgments:
More informationAdvection / Hyperbolic PDEs. PHY 604: Computational Methods in Physics and Astrophysics II
Advection / Hyperbolic PDEs Notes In addition to the slides and code examples, my notes on PDEs with the finite-volume method are up online: https://github.com/open-astrophysics-bookshelf/numerical_exercises
More informationFinite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
More informationNUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS
NUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS JOHN A. TRANGENSTEIN Department of Mathematics, Duke University Durham, NC 27708-0320 Ш CAMBRIDGE ЩР UNIVERSITY PRESS Contents 1 Introduction
More informationENO and WENO schemes. Further topics and time Integration
ENO and WENO schemes. Further topics and time Integration Tefa Kaisara CASA Seminar 29 November, 2006 Outline 1 Short review ENO/WENO 2 Further topics Subcell resolution Other building blocks 3 Time Integration
More informationNumerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation
More informationA New Fourth-Order Non-Oscillatory Central Scheme For Hyperbolic Conservation Laws
A New Fourth-Order Non-Oscillatory Central Scheme For Hyperbolic Conservation Laws A. A. I. Peer a,, A. Gopaul a, M. Z. Dauhoo a, M. Bhuruth a, a Department of Mathematics, University of Mauritius, Reduit,
More informationA Divergence-Free Upwind Code for Multidimensional Magnetohydrodynamic Flows 4
draft of August 0, 998 A Divergence-Free Upwind Code for Multidimensional Magnetohydrodynamic Flows 4 Dongsu Ryu, Francesco Miniati,T.W.Jones, and Adam Frank 3 ABSTRACT A description is given for preserving
More informationApproximate Harten-Lax-Van Leer (HLL) Riemann Solvers for Relativistic hydrodynamics and MHD
Approximate Harten-Lax-Van Leer (HLL) Riemann Solvers for Relativistic hydrodynamics and MHD Andrea Mignone Collaborators: G. Bodo, M. Ugliano Dipartimento di Fisica Generale, Universita di Torino (Italy)
More informationMath 660-Lecture 23: Gudonov s method and some theories for FVM schemes
Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationAn efficient implementation of the divergence free constraint in a discontinuous Galerkin method for magnetohydrodynamics on unstructured meshes
An efficient implementation of the divergence free constraint in a discontinuous Galerkin method for magnetohydrodynamics on unstructured meshes Christian Klingenberg, Frank Pörner, Yinhua Xia Abstract
More informationNUMERICAL METHODS IN ASTROPHYSICS An Introduction
-1 Series in Astronomy and Astrophysics NUMERICAL METHODS IN ASTROPHYSICS An Introduction Peter Bodenheimer University of California Santa Cruz, USA Gregory P. Laughlin University of California Santa Cruz,
More informationMATHEMATICAL ASPECTS OF NUMERICAL SOLUTION OF HYPERBOLIC SYSTEMS
K CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics I 18 MATHEMATICAL ASPECTS OF NUMERICAL SOLUTION OF HYPERBOLIC SYSTEMS ANDREI G. KULIKOVSKII NIKOLAI V. POGORELOV ANDREI YU. SEMENOV
More informationarxiv: v2 [math.na] 4 Mar 2014
Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics arxiv:1309.3344v [math.na] 4 Mar 014 Andrew J. Christlieb a, James A. Rossmanith
More informationThe PLUTO code for astrophysical gasdynamics
Mem. S.A.It. Suppl. Vol. 13, 67 c SAIt 009 Memorie della Supplementi The PLUTO code for astrophysical gasdynamics A. Mignone 1, 1 Dipartimento di Fisica Generale Amedeo Avogadro, Università degli Studi
More informationChp 4: Non-linear Conservation Laws; the Scalar Case. By Prof. Dinshaw S. Balsara
Chp 4: Non-linear Conservation Laws; the Scalar Case By Prof. Dinshaw S. Balsara 1 4.1) Introduction We have seen that monotonicity preserving reconstruction and iemann solvers are essential building blocks
More informationChapter 1 Direct Modeling for Computational Fluid Dynamics
Chapter 1 Direct Modeling for Computational Fluid Dynamics Computational fluid dynamics (CFD) is a scientific discipline, which aims to capture fluid motion in a discretized space. The description of the
More informationSpatial Discretization
Spatial Discretization Shengtai Li and Hui Li Los Alamos National Laboratory 1 Overview Finite Difference Method Finite Difference for Linear Advection Equation 3 Conservation Laws Modern Finite Difference
More informationCapSel Roe Roe solver.
CapSel Roe - 01 Roe solver keppens@rijnh.nl modern high resolution, shock-capturing schemes for Euler capitalize on known solution of the Riemann problem originally developed by Godunov always use conservative
More informationA Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations
A Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations Ch. Altmann, G. Gassner, F. Lörcher, C.-D. Munz Numerical Flow Models for Controlled
More informationOutline of lectures. Adding more physics. Doing Better: source terms in CTU unsplit integrator
Grid-based methods for hydrodynamics, MHD, and radiation hydrodynamics. Outline of lectures (Four lectures) Lecture 1. Introduction to physics and numerics Lecture 2. Operator split (ZEUS-like) methods
More informationFluid Dynamics. Part 2. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/17
Fluid Dynamics p.1/17 Fluid Dynamics Part 2 Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/17 Schemes Based on Flux-conservative Form By their very nature, the fluid equations
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationA recovery-assisted DG code for the compressible Navier-Stokes equations
A recovery-assisted DG code for the compressible Navier-Stokes equations January 6 th, 217 5 th International Workshop on High-Order CFD Methods Kissimmee, Florida Philip E. Johnson & Eric Johnsen Scientific
More informationOn divergence-free reconstruction schemes for CED and MHD
On divergence-free reconstruction schemes for CED and MHD Praveen Chandrashekar praveen@math.tifrbng.res.in Center for Applicable Mathematics Tata Institute of Fundamental Research Bangalore-560065, India
More informationA numerical study of SSP time integration methods for hyperbolic conservation laws
MATHEMATICAL COMMUNICATIONS 613 Math. Commun., Vol. 15, No., pp. 613-633 (010) A numerical study of SSP time integration methods for hyperbolic conservation laws Nelida Črnjarić Žic1,, Bojan Crnković 1
More informationPositivity-preserving high order schemes for convection dominated equations
Positivity-preserving high order schemes for convection dominated equations Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Xiangxiong Zhang; Yinhua Xia; Yulong Xing; Cheng
More informationInfo. No lecture on Thursday in a week (March 17) PSet back tonight
Lecture 0 8.086 Info No lecture on Thursday in a week (March 7) PSet back tonight Nonlinear transport & conservation laws What if transport becomes nonlinear? Remember: Nonlinear transport A first attempt
More informationApplying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models
0-0 Applying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models B. Srinivasan, U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle,
More informationRecent Progress in FLASH: High-Energy- Density Physics Applications
The Center for Astrophysical Thermonuclear Flashes Recent Progress in FLASH: High-Energy- Density Physics Applications Dongwook Lee The Physics of Intracluster Medium: Theory & Computation University of
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 59 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS The Finite Volume Method These slides are partially based on the recommended textbook: Culbert B.
More informationarxiv:astro-ph/ v1 7 Oct 2003
1 On the Divergence-Free Condition in Godunov-Type Schemes for Ideal Magnetohydrodynamics: the Upwind Constrained Transport Method arxiv:astro-ph/0310183v1 7 Oct 2003 P. Londrillo INAF - Osservatorio di
More informationX i t react. ~min i max i. R ij smallest. X j. Physical processes by characteristic timescale. largest. t diff ~ L2 D. t sound. ~ L a. t flow.
Physical processes by characteristic timescale Diffusive timescale t diff ~ L2 D largest Sound crossing timescale t sound ~ L a Flow timescale t flow ~ L u Free fall timescale Cooling timescale Reaction
More informationHyperbolic Divergence Cleaning for the MHD Equations
Journal of Computational Physics 175, 645 673 (2002) doi:10.1006/jcph.2001.6961, available online at http://www.idealibrary.com on Hyperbolic Divergence Cleaning for the MHD Equations A. Dedner, F. Kemm,
More informationThe one-dimensional equations for the fluid dynamics of a gas can be written in conservation form as follows:
Topic 7 Fluid Dynamics Lecture The Riemann Problem and Shock Tube Problem A simple one dimensional model of a gas was introduced by G.A. Sod, J. Computational Physics 7, 1 (1978), to test various algorithms
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationGas-Kinetic Relaxation (BGK-Type) Schemes for the Compressible Euler Equations
Gas-Kinetic Relaxation (BGK-Type) Schemes for the Compressible Euler Equations Iizln Xu* and Antony ~arneson~ Department of Mechanical and Aerospace Engineering Princeton University Princeton, New Jersey
More informationLES Simulations of Quiet Sun Magnetism
LES Simulations of Quiet Sun Magnetism Matthias Rempel HAO/NCAR Quiet sun magnetism Origin and spatial distribution of quiet sun field Small scale dynamo? Remnant field from large scale dynamo? Vögler,
More informationHigh-resolution finite volume methods for hyperbolic PDEs on manifolds
High-resolution finite volume methods for hyperbolic PDEs on manifolds Randall J. LeVeque Department of Applied Mathematics University of Washington Supported in part by NSF, DOE Overview High-resolution
More informationBound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu
Bound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu Division of Applied Mathematics Brown University Outline Introduction Maximum-principle-preserving for scalar conservation
More informationNumerical Simulations. Duncan Christie
Numerical Simulations Duncan Christie Motivation There isn t enough time to derive the necessary methods to do numerical simulations, but there is enough time to survey what methods and codes are available
More informationThe importance of including XMHD physics in HED codes
The importance of including XMHD physics in HED codes Charles E. Seyler, Laboratory of Plasma Studies, School of Electrical and Computer Engineering, Cornell University Collaborators: Nat Hamlin (Cornell)
More informationIdeal Magnetohydrodynamics (MHD)
Ideal Magnetohydrodynamics (MHD) Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 1, 2016 These lecture notes are largely based on Lectures in Magnetohydrodynamics
More informationImprovement of convergence to steady state solutions of Euler equations with. the WENO schemes. Abstract
Improvement of convergence to steady state solutions of Euler equations with the WENO schemes Shuhai Zhang, Shufen Jiang and Chi-Wang Shu 3 Abstract The convergence to steady state solutions of the Euler
More informationSung-Ik Sohn and Jun Yong Shin
Commun. Korean Math. Soc. 17 (2002), No. 1, pp. 103 120 A SECOND ORDER UPWIND METHOD FOR LINEAR HYPERBOLIC SYSTEMS Sung-Ik Sohn and Jun Yong Shin Abstract. A second order upwind method for linear hyperbolic
More informationA Fourth-Order Central Runge-Kutta Scheme for Hyperbolic Conservation Laws
A Fourth-Order Central Runge-Kutta Scheme for Hyperbolic Conservation Laws Mehdi Dehghan, Rooholah Jazlanian Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University
More informationICES REPORT A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws
ICES REPORT 7- August 7 A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws by Todd Arbogast, Chieh-Sen Huang, and Xikai Zhao The Institute for Computational Engineering and Sciences The
More informationConstrained Transport Method for the Finite Volume Evolution Galerkin Schemes with Application in Astrophysics
Project work at the Department of Mathematics, TUHH Constrained Transport Method for the Finite Volume Evolution Galerkin Schemes with Application in Astrophysics Katja Baumbach April 4, 005 Supervisor:
More informationSimple examples of MHD equilibria
Department of Physics Seminar. grade: Nuclear engineering Simple examples of MHD equilibria Author: Ingrid Vavtar Mentor: prof. ddr. Tomaž Gyergyek Ljubljana, 017 Summary: In this seminar paper I will
More informationDivergence- and Curl-Preserving Prolongation and Restriction Formulas
Journal of Computational Physics 180, 736 750 00) doi:10.1006/jcph.00.710 Divergence- and Curl-Preserving Prolongation and Restriction Formulas G. Tóth and P. L. Roe University of Michigan, Ann Arbor,
More information2.2. Methods for Obtaining FD Expressions. There are several methods, and we will look at a few:
.. Methods for Obtaining FD Expressions There are several methods, and we will look at a few: ) Taylor series expansion the most common, but purely mathematical. ) Polynomial fitting or interpolation the
More informationarxiv: v4 [astro-ph.im] 27 Apr 2011
A ROBUST NUMERICAL SCHEME FOR HIGHLY COMPRESSIBLE MAGNETOHYDRODYNAMICS: NONLINEAR STABILITY, IMPLEMENTATION AND TESTS K. WAAGAN, C. FEDERRATH, AND C. KLINGENBERG arxiv:1101.3007v4 [astro-ph.im] 27 Apr
More informationPiecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation
Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation Sergey D. Ustyugov, Mikhail V. Popov, Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047, Moscow,
More informationRESEARCH HIGHLIGHTS. WAF: Weighted Average Flux Method
RESEARCH HIGHLIGHTS (Last update: 3 rd April 2013) Here I briefly describe my contributions to research on numerical methods for hyperbolic balance laws that, in my view, have made an impact in the scientific
More informationDivergence Formulation of Source Term
Preprint accepted for publication in Journal of Computational Physics, 2012 http://dx.doi.org/10.1016/j.jcp.2012.05.032 Divergence Formulation of Source Term Hiroaki Nishikawa National Institute of Aerospace,
More informationShock-capturing Schemes for a Collisionless Two-fluid Plasma Model
Shock-capturing Schemes for a Collisionless Two-fluid Plasma Model E. Alec Johnson Department of Mathematics, UW-Madison Presented on August 29, 2 at Sandia National Laboratories c 23, Kenneth Lang, Tufts
More informationImproving Dynamical Core Scalability, Accuracy, and Limi:ng Flexibility with the ADER- DT Time Discre:za:on
Improving Dynamical Core Scalability, Accuracy, and Limi:ng Flexibility with the ADER- DT Time Discre:za:on Matthew R. Norman Scientific Computing Group National Center for Computational Sciences Oak Ridge
More informationarxiv: v3 [physics.comp-ph] 22 Mar 2018
Preprint 3rd March 08 Compiled using MNRAS LATEX style file v3.0 arxiv:504.0985v3 [physics.comp-ph] Mar 08 Fourth Order Accurate Finite Volume CWENO Scheme For Astrophysical MHD Problems Prabal Singh Verma,,4,
More informationA NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS
A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS HASEENA AHMED AND HAILIANG LIU Abstract. High resolution finite difference methods
More informationPLASMA ASTROPHYSICS. ElisaBete M. de Gouveia Dal Pino IAG-USP. NOTES: (references therein)
PLASMA ASTROPHYSICS ElisaBete M. de Gouveia Dal Pino IAG-USP NOTES:http://www.astro.iag.usp.br/~dalpino (references therein) ICTP-SAIFR, October 7-18, 2013 Contents What is plasma? Why plasmas in astrophysics?
More informationIntroduction to Magnetohydrodynamics (MHD)
Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection Aim Derivation of MHD equations from conservation laws Quasi-neutrality
More informationReduced MHD. Nick Murphy. Harvard-Smithsonian Center for Astrophysics. Astronomy 253: Plasma Astrophysics. February 19, 2014
Reduced MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 19, 2014 These lecture notes are largely based on Lectures in Magnetohydrodynamics by Dalton
More informationComputational Fluid Dynamics. PHY 688: Numerical Methods for (Astro)Physics
Computational Fluid Dynamics Hydrodynamics When we discussed PDEs, we focused so far on scalar PDEs Often we wish to study systems of PDEs. Here we'll look at the equations of hydrodynamics Nonlinear system
More informationRecapitulation: Questions on Chaps. 1 and 2 #A
Recapitulation: Questions on Chaps. 1 and 2 #A Chapter 1. Introduction What is the importance of plasma physics? How are plasmas confined in the laboratory and in nature? Why are plasmas important in astrophysics?
More informationArt Checklist. Journal Code: Article No: 6959
Art Checklist Journal Code: JCPH Article No: 6959 Disk Recd Disk Disk Usable Art # Y/N Format Y/N Remarks Fig. 1 Y PS Y Fig. 2 Y PS Y Fig. 3 Y PS Y Fig. 4 Y PS Y Fig. 5 Y PS Y Fig. 6 Y PS Y Fig. 7 Y PS
More informationMacroscopic plasma description
Macroscopic plasma description Macroscopic plasma theories are fluid theories at different levels single fluid (magnetohydrodynamics MHD) two-fluid (multifluid, separate equations for electron and ion
More informationFinite volumes for complex applications In this paper, we study finite-volume methods for balance laws. In particular, we focus on Godunov-type centra
Semi-discrete central schemes for balance laws. Application to the Broadwell model. Alexander Kurganov * *Department of Mathematics, Tulane University, 683 St. Charles Ave., New Orleans, LA 708, USA kurganov@math.tulane.edu
More informationLocally Divergence-Free Central Discontinuous Galerkin Methods for Ideal MHD Equations
Locally Divergence-Free Central Discontinuous Galerkin Methods for Ideal MHD Equations Sergey Yakovlev, Liwei Xu and Fengyan Li Abstract In this paper, we propose and numerically investigate a family of
More informationVISCOUS FLUX LIMITERS
VISCOUS FLUX LIMITERS E. F. Toro Department of Aerospace Science College of Aeronautics Cranfield Institute of Technology Cranfield, Beds MK43 OAL England. Abstract We present Numerical Viscosity Functions,
More informationA Scalable, Parallel Implementation of Weighted, Non-Linear Compact Schemes
A Scalable, Parallel Implementation of Weighted, Non-Linear Compact Schemes Debojyoti Ghosh Emil M. Constantinescu Jed Brown Mathematics Computer Science Argonne National Laboratory SIAM Annual Meeting
More informationHybrid modeling of plasmas
Hybrid modeling of plasmas Mats Holmström Swedish Institute of Space Physics Kiruna, Sweden ENUMATH 2009 Uppsala, June 30 matsh@irf.se www.irf.se/~matsh/ Outline Background and motivation Space plasma
More informationDivergence-free interpolation of vector fields from point values exact B = 0 in numerical simulations
Mon. Not. R. Astron. Soc. 413, L76 L80 (2011) doi:10.1111/j.1745-3933.2011.01037.x Divergence-free interpolation of vector fields from point values exact B = 0 in numerical simulations Colin P. McNally
More informationFUNDAMENTALS OF LAX-WENDROFF TYPE APPROACH TO HYPERBOLIC PROBLEMS WITH DISCONTINUITIES
6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational Fluid Dynamics (ECFD 7) 1115 June 018, Glasgow, UK FUNDAMENTALS OF LAX-WENDROFF TYPE APPROACH TO HYPERBOLIC
More informationPhysical Diffusion Cures the Carbuncle Phenomenon
Physical Diffusion Cures the Carbuncle Phenomenon J. M. Powers 1, J. Bruns 1, A. Jemcov 1 1 Department of Aerospace and Mechanical Engineering University of Notre Dame, USA Fifty-Third AIAA Aerospace Sciences
More informationFinite Volume for Fusion Simulations
Finite Volume for Fusion Simulations Elise Estibals, Hervé Guillard, Afeintou Sangam To cite this version: Elise Estibals, Hervé Guillard, Afeintou Sangam. Finite Volume for Fusion Simulations. Jorek Meeting
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationQuest for Osher-Type Riemann Solver
p./7 Quest for Osher-Type Riemann Solver for Ideal MHD Equations Project Proposal EMRAS 2 Hervé Guillard, arry Koren, oniface Nkonga IRM, July 9 August 27, 2 p.2/7 Wine guide to Riemann solvers Godunov:
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationMultigrid solvers for equations arising in implicit MHD simulations
Multigrid solvers for equations arising in implicit MHD simulations smoothing Finest Grid Mark F. Adams Department of Applied Physics & Applied Mathematics Columbia University Ravi Samtaney PPPL Achi Brandt
More informationNumerical solutions of magnetohydrodynamic equations
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 59, No., 11 DOI: 1.478/v1175-11-7-9 Numerical solutions of magnetohydrodynamic equations K. MURAWSKI Faculty of Mathematics, Physics
More informationStable Numerical Scheme for the Magnetic Induction Equation with Hall Effect
Stable Numerical Scheme for the Magnetic Induction Equation with Hall Effect Paolo Corti joint work with Siddhartha Mishra ETH Zurich, Seminar for Applied Mathematics 17-19th August 2011, Pro*Doc Retreat,
More informationAn Improved Non-linear Weights for Seventh-Order WENO Scheme
An Improved Non-linear Weights for Seventh-Order WENO Scheme arxiv:6.06755v [math.na] Nov 06 Samala Rathan, G Naga Raju Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur,
More information