Strong Stability-Preserving (SSP) High-Order Time Discretization Methods
|
|
- Edmund Dean
- 6 years ago
- Views:
Transcription
1 Strong Stability-Preserving (SSP) High-Order Time Discretization Methods Xinghui Zhong 12/09/ 2009
2 Outline 1 Introduction Why SSP methods Idea History/main reference 2 Explicit SSP Runge-Kutta Methods SSP Theory Optimal SSPRK Methods for Nonlinear Problems Optimal SSPRK methods for Linear Operator Low Storage Methods 3 Explicit SSP Multi Step Methods SSP Theory Order Barriers 4 Implicit SSP Methods Diagonally Implicit Runge-Kutta methods Implicit SSP Multi Step Methods 5 Summary
3 Why SSP methods Time-dependent PDE = ODE Lax equivalence theorem: A linear method consistent with a linear problem stability convergence. Strang s theorem For nonlinear problems with sufficiently smooth solution, if an approximation is consistent and its linearized version is L 2 stable, = convergence problems with discontinuous solutions??? high order spatial discretization + forward Euler time stepping method
4 Why SSP methods hyperbolic conservation law u t = f (u) x. ODE u t = L(u) L(u)+ forward Euler: stability properties L(u)+ high order time discretization??
5 Why SSP methods Burger s equation Initial condition ( ) u 2 u t + 2 x = 0. u(x, 0) = { 1, if x 0, 0.5, if x > 0, Spatial discretization 2nd order minmod based Monotone Upstream-centered Schemes for Conservation Law (MUSCL)
6 Why SSP methods Time discretization SSP 2nd order RK non-ssp method u (1) = u n + tl(u n ) u n+1 = 1 2 un u(1) tl(u(1) ). u (1) = u n 20 tl(u n ) u n+1 = u n u(1) 1 40 tl(u(1) ).
7 Idea Idea Assume first order forward Euler time discretization of the method of lines ODE is strongly stable under, when t t FE, and then try to find a higher order time discretization that maintains the strong stability for the same norm, under t t. strong stability A sequence u n is said to be strongly stable in a given norm, provided u n+1 u n total variation diminishing (TVD) property TV (u n+1 ) TV (u n ) where TV (u) = j u j+1 u j.
8 History/main reference 1988 C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, II 1988 C.-W. Shu, Total-Variation-Diminishing Time Discretizations 1998 S. Gottlieb and C.-W. Shu Total-Variation-Diminishing Runge-Kutta Schemes 2001 S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong Stability Preserving High-Order Time Discretization Methods 2002 S. J. Ruuth and R. J. Spiteri. Two Barriers on Strong-Stability-Preserving Time Discretization Methods 2005 S. Gottlieb. On High Order Strong Stability Preserving Runge-Kutta and Multi Step Time Discretizations. J.S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-Dependent Problems
9 SSP Theory A general m-stage Runge-Kutta method u (0) = u n, i 1 ( ) u (i) = α i,k u (k) + tβ i,k L(u (k) ), α i,k 0, i = 1,, m, k=0 u n+1 = u (m). Consistency = i 1 k=0 α i,k = 1 β i,k 0, t β i,k α i,k t β i,k < 0, L is replace by L t β i,k α i,k t L approximates the same spatial derivative as L Strong stability property where u n+1 u n u n+1 = u n t L(u n ).
10 SSP Theory Theory (C.-W. Shu and S. Osher) If under CFL restriction and if u n + t L(u n ) u n t t FE, (1) u n t L(u n ) u n under the CFL restriction (1), Then the RK method is SSP u n+1 u n, under the CFL restriction, t c t FE, c = min i,k α i,k β i,k Provided L is replaced by L whenever β i,k is negative.
11 Optimal SSPRK Methods for Nonlinear Problems Optimal c: as large as possible L and L: avoid negative β i,k whenever possible definition effective CFL c eff = c l, where c: CFL coefficient l: the number of computations of L and L required per time step.
12 Optimal SSPRK Methods for Nonlinear Problems SSPRK (2,2): If we require β i,k 0, then u (1) = u n + t L(u n ) u n+1 = 1 2 un u(1) t L(u(1) ). with c = 1 and c eff = 1/2. SSPRK (3,3): If we require β i,k 0, then u (1) = u n + tl(u n ) u (2) = 3 4 un u(1) t L(u(1) ) u n+1 = 1 3 un u(2) t L(u(2) ) with c = 1 and c eff = 1/3. Shu-Osher method
13 Optimal SSPRK Methods for Nonlinear Problems Proposition (S.Gottlieb and C.-W. Shu) The four-stage, fourth-order SSP Runge-Kutta scheme with a nonzero CFL coefficient c must have at least one negative β i,k. Spiteri and Ruuth proved that any SSPRK with nonzero CFL of order p > 4 will have negative β i,k. Spiteri and Ruuth developed fourth order methods with m = 5, 6, 7 and 8 stages.
14 Optimal SSPRK Methods for Nonlinear Problems SSPRK(5,4) u (1) =u n t L(u n ), u (2) = u n u (1) tl(u (1) ), u (3) = u n u (2) t L(u (2) ), u (4) = u n u (3) t L(u (3) ), u n+1 = u (2) u (3) t L(u (3) u (4) t L(u (4) ) with c = and c eff =
15 Optimal SSPRK methods for Linear Operator Denote If L is a linear constant coefficient operator, then L(u) = L u. Theory (Spiteri and Ruuth) Consider SSPRK (m,p) methods with α i,k, β i,k 0 applied to u t = L u. The CFL restriction then satisfies t (m p + 1) t FE.
16 Optimal SSPRK methods for Linear Operator Table 1. Optimal CFL coefficients c, and the Corresponding Effective CFL c eff of SSPRK linear (m, p)
17 Optimal SSPRK methods for Linear Operator SSPRK linear (m,m) u (i) = u (i 1) + t Lu (i 1), i = 1,, m 1, u (m) = m 2 k=0 where α 1,0 = 1 and α m,k u (k) + α m,m 1 (u (m 1) + t L u (m 1)), α m,k = 1 k α m 1,k 1, k = 1,, m 2, α m,m 1 = 1 m!, m 1 α m,0 = 1 α m,k. k=1 with c = 1 and c eff = 1/m.
18 Optimal SSPRK methods for Linear Operator Table 2. Coefficients α m,j of SSPRK linear (m, m)
19 Optimal SSPRK methods for Linear Operator SSPRK linear (m, 1) ( u (i) = 1 + t ) m L u (i 1), i = 1,, m. with c = m and c eff = 1. SSPRK linear (m, 2) ( u (i) = 1 + t ) m 1 L u (i 1), i = 1,, m 1, u (m) = 1 m u(0) + m 1 ( 1 + t ) m m 1 L u (m 1), with c = m 1 and c eff = (m 1)/m.
20 Low Storage Methods The general low-storage RK methods: u (0) = u n, k i = A i k i 1 + t L(u (i 1) ), i = 1,, m u (i) = u (i 1) + B i k i, i = 1,, m, B 1 = c. u n+1 = u (m). M. Carpenter and C. Kennedy Fourth-order 2N-storage Runge-Kutta schemes, all the low-storage RK (3, 3). the best SSP 3rd order with c = 0.92 and c eff = 0.32(S. Gottlieb and C.-W. Shu). less optimal than SSPRK (3,3) useful for large-scale calculations classes of the low-storage RK (5, 4) unable to find SSP methods with β i,k > 0.
21 The Need for SSP Property in the Intermediate Stages Remark SSPRK methods have also intermediate stage SSP properties. u (i) u (i 1), i = 1, m Consider u t u x = 0, 0 x 1 { 0 if x 1 u(0, x) = 2, 1 if x > 1 2. Spatial discretization u t = L(u) = u(t, x j+1) u(t, x j ) x
22 Time discretization SSPRK (2, 2) non SSPRK u (1) = u n + tl(u n ) u (1) = u n 20 tl(u n ) u n+1 = 1 2 un u(1) u n+1 = u n u(1) tl(u(1) ) tl(u(1) ). Figure: Intermediate stage solution u (1) after 10 time steps.
23 SSP Theory Explicit SSP multi step methods m ( ) u n+1 = α i u n+1 i + tβ i L(u n+1 i ), α i 0. i=1 Theory (S. Gottlieb, C.-W. Shu and E. Tadmor) If u n + t L(u n ) u n under CFL restriction t t FE, (2) and if u n t L(u n ) u n under the CFL restriction (2), Then the multi step method is SSP u n+1 u n, under the CFL restriction, t c t FE, c = min i,k α i β i Provided L is replaced by L whenever β i is negative.
24 Order Barriers The explicit SSP multi step methods are p-th order accurate if m α i = 1, i=1 ( m m ) i k α i = k i k 1 β i.k = 1,, p. i=1 i=1 Proposition (S. Gottlieb, C.-W. Shu and E. Tadmor) For m 2, there is no m-step, mth-order SSP method with all nonnegative β i, and there is no m-step SSP method of order (m + 1).
25 Diagonally Implicit Runge-Kutta methods DIRK method u (0) = u n i 1 u (i) = α i,k u (k) + tβ i L(u (i) ), α i,k 0, i = 1,, m, k=0 u n+1 = u (m). Remark β i < 0, introduce an associated operator L approximate the same spatial derivative(s) as L unconditionally strong stable for first-order implicit Euler, backward in time: u n+1 = u n t L(u n+1 ).
26 Diagonally Implicit Runge-Kutta methods If u n+1 = u n + t L(u n+1 ) and u n+1 = u n t L(u n+1 ) is unconditionally strong stable, u n+1 u n Then the above DIRK methods are unconditionally strong stable under the same norm. Provided L is replaced by L whenever β i is negative. Proposition (S.Gottlieb, C.-W. Shu and E. Tadmor) If the above DIRK is at least second-order accurate, then α i,k cannot be all nonnegative.
27 Implicit SSP Multi Step Methods Implicit SSP multi step methods u n+1 = m α i u n+1 i + tβ 0 L(u n+1 ). i=1 If u n+1 = u n + t L(u n+1 ) and u n+1 = u n t L(u n+1 ) is unconditionally strong stable, u n+1 u n. Then this method would be unconditionally strong stable under the same norm Provided L is replaced by L whenever β 0 is negative. Proposition (S.Gottlieb, C.-W. Shu and E. Tadmor) If the above multi-step method is at least second-order accurate, then α i cannot be all nonnegative.
28 Summary SSP methods preserves the strong stability, in any norm or semi-norm, of the forward Euler (for explicit methods) or the backward Euler (for implicit methods). SSP methods are very useful for method of lines numerical schemes for PDEs, especially in solving hyperbolic problems. The goal to design higher order implicit SSP methods, which share the strong stability properties of implicit Euler, without any restriction on the time step, cannot be realized.
29 Thank you
On High Order Strong Stability Preserving Runge Kutta and Multi Step Time Discretizations
Journal of Scientific Computing, Vol. 5, Nos. /, November 005 ( 005) DOI: 0.007/s095-00-65-5 On High Order Strong Stability Preserving Runge Kutta and Multi Step Time Discretizations Sigal Gottlieb Received
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 informationStrong Stability Preserving Properties of Runge Kutta Time Discretization Methods for Linear Constant Coefficient Operators
Journal of Scientific Computing, Vol. 8, No., February 3 ( 3) Strong Stability Preserving Properties of Runge Kutta Time Discretization Methods for Linear Constant Coefficient Operators Sigal Gottlieb
More informationStrong Stability Preserving Time Discretizations
AJ80 Strong Stability Preserving Time Discretizations Sigal Gottlieb University of Massachusetts Dartmouth Center for Scientific Computing and Visualization Research November 20, 2014 November 20, 2014
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 informationTOTAL VARIATION DIMINISHING RUNGE-KUTTA SCHEMES
MATHEMATICS OF COMPUTATION Volume 67 Number 221 January 1998 Pages 73 85 S 0025-5718(98)00913-2 TOTAL VARIATION DIMINISHING RUNGE-KUTTA SCHEMES SIGAL GOTTLIEB AND CHI-WANG SHU Abstract. In this paper we
More informationA New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods Raymond J. Spiteri Steven J. Ruuth Technical Report CS-- May 6, Faculty of Computer Science 65 University Ave.,
More informationDesign of optimal Runge-Kutta methods
Design of optimal Runge-Kutta methods David I. Ketcheson King Abdullah University of Science & Technology (KAUST) D. Ketcheson (KAUST) 1 / 36 Acknowledgments Some parts of this are joint work with: Aron
More informationARTICLE IN PRESS Mathematical and Computer Modelling ( )
Mathematical and Computer Modelling Contents lists available at ScienceDirect Mathematical and Computer Modelling ournal homepage: wwwelseviercom/locate/mcm Total variation diminishing nonstandard finite
More informationA Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions.
A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions. Zachary Grant 1, Sigal Gottlieb 2, David C. Seal 3 1 Department of
More informationOptimal Implicit Strong Stability Preserving Runge Kutta Methods
Optimal Implicit Strong Stability Preserving Runge Kutta Methods David I. Ketcheson, Colin B. Macdonald, Sigal Gottlieb. February 21, 2008 Abstract Strong stability preserving (SSP) time discretizations
More informationOptimal Implicit Strong Stability Preserving Runge Kutta Methods
Optimal Implicit Strong Stability Preserving Runge Kutta Methods David I. Ketcheson, Colin B. Macdonald, Sigal Gottlieb. August 3, 2007 Abstract Strong stability preserving (SSP) time discretizations were
More informationStrong Stability Preserving High-order Time Discretization Methods
NASA/CR-000-009 ICASE Report No. 000-5 Strong Stability Preserving High-order Time Discretization Methods Sigal Gottlieb University of Massachusetts, Dartmouth, Massachusetts Chi-Wang Shu Brown University,
More informationc 2013 Society for Industrial and Applied Mathematics
SIAM J. NUMER. ANAL. Vol. 5, No. 4, pp. 249 265 c 203 Society for Industrial and Applied Mathematics STRONG STABILITY PRESERVING EXPLICIT RUNGE KUTTA METHODS OF MAXIMAL EFFECTIVE ORDER YIANNIS HADJIMICHAEL,
More informationABSTRACT. In this paper we review and further develop a class of strong stability preserving (SSP)
........................ c000 00 Strong Stability Preserving High Order Time Discretization Methods Sigal Gottlieb Chi-Wang Shu y and Eitan Tadmor z ABSTRACT In this paper we review and further develop
More informationSurprising Computations
.... Surprising Computations Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca www.cs.ubc.ca/ ascher/ Uri Ascher (UBC) Surprising Computations Fall 2012 1 / 67 Motivation.
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 informationA Fifth Order Flux Implicit WENO Method
A Fifth Order Flux Implicit WENO Method Sigal Gottlieb and Julia S. Mullen and Steven J. Ruuth April 3, 25 Keywords: implicit, weighted essentially non-oscillatory, time-discretizations. Abstract The weighted
More informationStepsize Restrictions for Boundedness and Monotonicity of Multistep Methods
Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods W. Hundsdorfer, A. Mozartova, M.N. Spijker Abstract In this paper nonlinear monotonicity and boundedness properties are analyzed
More informationStrong Stability of Singly-Diagonally-Implicit Runge-Kutta Methods
Strong Stability of Singly-Diagonally-Implicit Runge-Kutta Methods L. Ferracina and M. N. Spijker 2007, June 4 Abstract. This paper deals with the numerical solution of initial value problems, for systems
More informationIntroduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions
More information30 crete maximum principle, which all imply the bound-preserving property. But most
3 4 7 8 9 3 4 7 A HIGH ORDER ACCURATE BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME FOR SCALAR CONVECTION DIFFUSION EQUATIONS HAO LI, SHUSEN XIE, AND XIANGXIONG ZHANG Abstract We show that the classical
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationNumerical Methods for the Optimal Control of Scalar Conservation Laws
Numerical Methods for the Optimal Control of Scalar Conservation Laws Sonja Steffensen, Michael Herty, and Lorenzo Pareschi RWTH Aachen University, Templergraben 55, D-52065 Aachen, GERMANY {herty,steffensen}@mathc.rwth-aachen.de
More informationKrylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations
mathematics Article Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations Michael Machen and Yong-Tao Zhang * Department of Applied and Computational Mathematics and Statistics,
More informationA Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations
A Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations Hao Li Math Dept, Purdue Univeristy Ocean University of China, December, 2017 Joint work with
More informationHigh Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation
High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation Faheem Ahmed, Fareed Ahmed, Yongheng Guo, Yong Yang Abstract This paper deals with
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 informationA parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows.
A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows Tao Xiong Jing-ei Qiu Zhengfu Xu 3 Abstract In Xu [] a class of
More informationTHE CLOSEST POINT METHOD FOR TIME-DEPENDENT PROCESSES ON SURFACES
THE CLOSEST POINT METHOD FOR TIME-DEPENDENT PROCESSES ON SURFACES by Colin B. Macdonald B.Sc., Acadia University, 200 M.Sc., Simon Fraser University, 2003 a thesis submitted in partial fulfillment of the
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 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 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 informationNumerical Oscillations and how to avoid them
Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A.
More informationTutorial 2. Introduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2
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 informationGradient Method Based on Roots of A
Journal of Scientific Computing, Vol. 5, No. 4, 000 Solving Ax =b Using a Modified Conjugate Gradient Method Based on Roots of A Paul F. Fischer and Sigal Gottlieb Received January 3, 00; accepted February
More informationAn Assessment of Semi-Discrete Central Schemes for Hyperbolic Conservation Laws
SANDIA REPORT SAND2003-3238 Unlimited Release Printed September 2003 An Assessment of Semi-Discrete Central Schemes for Hyperbolic Conservation Laws Mark A. Christon David I. Ketcheson Allen C. Robinson
More informationA High Order WENO Scheme for a Hierarchical Size-Structured Model. Abstract
A High Order WENO Scheme for a Hierarchical Size-Structured Model Jun Shen 1, Chi-Wang Shu 2 and Mengping Zhang 3 Abstract In this paper we develop a high order explicit finite difference weighted essentially
More informationRECENT DEVELOPMENTS IN COMPUTATIONAL REACTOR ANALYSIS
RECENT DEVELOPMENTS IN COMPUTATIONAL REACTOR ANALYSIS Dean Wang April 30, 2015 24.505 Nuclear Reactor Physics Outline 2 Introduction and Background Coupled T-H/Neutronics Safety Analysis Numerical schemes
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 informationConstructing High-Order Runge-Kutta Methods with Embedded Strong-Stability-Preserving Pairs
Constructing High-Order Runge-Kutta Methods with Embedded Strong-Stability-Preserving Pairs by Colin Barr Macdonald B.Sc., Acadia University, 200 a thesis submitted in partial fulfillment of the requirements
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 informationEntropy stable high order discontinuous Galerkin methods. for hyperbolic conservation laws
Entropy stable high order discontinuous Galerkin methods for hyperbolic conservation laws Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Tianheng Chen, and with Yong Liu
More informationRunge Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing, Vol. 16, No. 3, September 2001 ( 2001) Review Article Runge Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Bernardo Cockburn 1 and Chi-Wang Shu
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 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 informationLast time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler:
Lecture 7 18.086 Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler: U j,n+1 t U j,n = U j+1,n 2U j,n + U j 1,n x 2 Expected accuracy: O(Δt) in time,
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 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 minimum entropy principle of high order schemes for gas dynamics. equations 1. Abstract
A minimum entropy principle of high order schemes for gas dynamics equations iangxiong Zhang and Chi-Wang Shu 3 Abstract The entropy solutions of the compressible Euler equations satisfy a minimum principle
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 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 informationStrong Stability Preserving High-order Time Discretization Methods
NASA/CR-2000-20093 ICASE Report No. 2000-5 Strong Stability Preserving High-order Time Discretization Methods Sigal Gottlieb University of Massachusetts, Dartmouth, Massachusetts Chi-WangShu Brown University,
More informationCONSTRUCTION OF HIGH-ORDER ADAPTIVE IMPLICIT METHODS FOR RESERVOIR SIMULATION
CONSTRUCTION OF HIGH-ORDER ADAPTIVE IMPLICIT METHODS FOR RESERVOIR SIMULATION A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
More informationRunge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter
Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter Jun Zhu, inghui Zhong, Chi-Wang Shu 3 and Jianxian Qiu 4 Abstract In this paper, we propose a new type of weighted
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 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 informationEfficient time discretization for local discontinuous Galerkin methods
Efficient time discretization for local discontinuous Galerkin methods Yinhua Xia, Yan Xu and Chi-Wang Shu Abstract In this paper, we explore a few efficient time discretization techniques for the local
More informationON MONOTONICITY AND BOUNDEDNESS PROPERTIES OF LINEAR MULTISTEP METHODS
MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 655 672 S 0025-578(05)0794- Article electronically published on November 7, 2005 ON MONOTONICITY AND BOUNDEDNESS PROPERTIES OF LINEAR MULTISTEP METHODS
More informationDedicated to the 70th birthday of Professor Lin Qun
Journal of Computational Mathematics, Vol.4, No.3, 6, 39 5. ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT ) Zhengfu Xu (Department of Mathematics, Pennsylvania
More informationAN OPTIMALLY ACCURATE SPECTRAL VOLUME FORMULATION WITH SYMMETRY PRESERVATION
AN OPTIMALLY ACCURATE SPECTRAL VOLUME FORMULATION WITH SYMMETRY PRESERVATION Fareed Hussain Mangi*, Umair Ali Khan**, Intesab Hussain Sadhayo**, Rameez Akbar Talani***, Asif Ali Memon* ABSTRACT High order
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationApproximation des systemes hyperboliques par elements finis continus non uniformes en dimension quelconque
Approximation des systemes hyperboliques par elements finis continus non uniformes en dimension quelconque Jean-Luc Guermond and Bojan Popov Department of Mathematics Texas A&M University Séminaire du
More informationA second-order asymptotic-preserving and positive-preserving discontinuous. Galerkin scheme for the Kerr-Debye model. Abstract
A second-order asymptotic-preserving and positive-preserving discontinuous Galerkin scheme for the Kerr-Debye model Juntao Huang 1 and Chi-Wang Shu Abstract In this paper, we develop a second-order asymptotic-preserving
More informationFourier analysis for discontinuous Galerkin and related methods. Abstract
Fourier analysis for discontinuous Galerkin and related methods Mengping Zhang and Chi-Wang Shu Abstract In this paper we review a series of recent work on using a Fourier analysis technique to study the
More informationInverse Lax-Wendroff Procedure for Numerical Boundary Conditions of. Conservation Laws 1. Abstract
Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Conservation Laws Sirui Tan and Chi-Wang Shu 3 Abstract We develop a high order finite difference numerical boundary condition for solving
More informationResearch Article Solution of the Porous Media Equation by a Compact Finite Difference Method
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2009, Article ID 9254, 3 pages doi:0.55/2009/9254 Research Article Solution of the Porous Media Equation by a Compact Finite Difference
More informationA Stable Spectral Difference Method for Triangles
A Stable Spectral Difference Method for Triangles Aravind Balan 1, Georg May 1, and Joachim Schöberl 2 1 AICES Graduate School, RWTH Aachen, Germany 2 Institute for Analysis and Scientific Computing, Vienna
More informationTime stepping methods
Time stepping methods ATHENS course: Introduction into Finite Elements Delft Institute of Applied Mathematics, TU Delft Matthias Möller (m.moller@tudelft.nl) 19 November 2014 M. Möller (DIAM@TUDelft) Time
More informationNumerische Mathematik
Numer. Math. ) :545 563 DOI.7/s--443-7 Numerische Mathematik A minimum entropy principle of high order schemes for gas dynamics equations iangxiong Zhang Chi-Wang Shu Received: 7 July / Revised: 5 September
More informationDeutscher Wetterdienst
Stability Analysis of the Runge-Kutta Time Integration Schemes for the Convection resolving Model COSMO-DE (LMK) COSMO User Seminar, Langen 03.+04. March 2008 Michael Baldauf Deutscher Wetterdienst, Offenbach,
More informationScalable Non-Linear Compact Schemes
Scalable Non-Linear Compact Schemes Debojyoti Ghosh Emil M. Constantinescu Jed Brown Mathematics Computer Science Argonne National Laboratory International Conference on Spectral and High Order Methods
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 informationMA/CS 615 Spring 2019 Homework #2
MA/CS 615 Spring 019 Homework # Due before class starts on Feb 1. Late homework will not be given any credit. Collaboration is OK but not encouraged. Indicate on your report whether you have collaborated
More informationLie Algebras and Burger s Equation: A Total Variation Diminishing Method on Manifold
Applied Mathematical Sciences, Vol. 11, 2017, no. 27, 1313-1325 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.74127 Lie Algebras and Burger s Equation: A Total Variation Diminishing Method
More informationMethod of Lines. Received April 20, 2009; accepted July 9, 2009
Method of Lines Samir Hamdi, William E. Schiesser and Graham W. Griffiths * Ecole Polytechnique, France; Lehigh University, USA; City University,UK. Received April 20, 2009; accepted July 9, 2009 The method
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 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 informationEntropic Schemes for Conservation Laws
CONSTRUCTVE FUNCTON THEORY, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2002, pp. 1-6. Entropic Schemes for Conservation Laws Bojan Popov A new class of Godunov-type numerical methods (called here entropic)
More informationImprovements of Unsteady Simulations for Compressible Navier Stokes Based on a RK/Implicit Smoother Scheme
Improvements of Unsteady Simulations for Compressible Navier Stokes Based on a RK/Implicit Smoother Scheme Oren Peles and Eli Turkel Department of Applied Mathematics, Tel-Aviv University In memoriam of
More informationStrong stability preserving high order time discretizations.
SIGAL GOTTLIEB Mathematics Department 285 Old Westport Road North Dartmouth, MA 02747 sgottlieb@umassd.edu 322 Cole Avenue Providence, RI 02906 Phone: (401) 751-9416 sigalgottlieb@yahoo.com Current Research
More informationA class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes
Science in China Series A: Mathematics Aug., 008, Vol. 51, No. 8, 1549 1560 www.scichina.com math.scichina.com www.springerlink.com A class of the fourth order finite volume Hermite weighted essentially
More informationYINGJIE LIU, CHI-WANG SHU, EITAN TADMOR, AND MENGPING ZHANG
CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NON-OSCILLATORY HIERARCHICAL RECONSTRUCTION YINGJIE LIU, CHI-WANG SHU, EITAN TADMOR, AND MENGPING ZHANG Abstract. The central scheme of
More informationAnti-diffusive finite difference WENO methods for shallow water with. transport of pollutant
Anti-diffusive finite difference WENO methods for shallow water with transport of pollutant Zhengfu Xu 1 and Chi-Wang Shu 2 Dedicated to Professor Qun Lin on the occasion of his 70th birthday Abstract
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme
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 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 informationSelected HW Solutions
Selected HW Solutions HW1 1 & See web page notes Derivative Approximations. For example: df f i+1 f i 1 = dx h i 1 f i + hf i + h h f i + h3 6 f i + f i + h 6 f i + 3 a realmax 17 1.7014 10 38 b realmin
More informationNumerical resolution of discontinuous Galerkin methods for time dependent. wave equations 1. Abstract
Numerical resolution of discontinuous Galerkin methods for time dependent wave equations Xinghui Zhong 2 and Chi-Wang Shu Abstract The discontinuous Galerkin DG method is known to provide good wave resolution
More informationStability of the fourth order Runge-Kutta method for time-dependent partial. differential equations 1. Abstract
Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations 1 Zheng Sun 2 and Chi-Wang Shu 3 Abstract In this paper, we analyze the stability of the fourth order
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 29 Almost Done! No
More informationSolution of Two-Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers
Solution of Two-Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers Alexander Kurganov, 1, * Eitan Tadmor 2 1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan
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 informationAdaptive WENO Schemes for Singular in Space and Time Solutions of Nonlinear Degenerate Reaction-Diffusion Problems
EPJ Web of Conferences 108, 0019 (016) DOI: 10.1051/ epjconf/ 0161080019 C Owned by the authors, published by EDP Sciences, 016 Adaptive WENO Schemes for Singular in Space and Time Solutions of Nonlinear
More informationarxiv: v2 [math.na] 24 Mar 2016
arxiv:1504.04107v2 [math.na] 24 Mar 2016 Strong stability preserving explicit linear multistep methods with variable step size Yiannis Hadjimichael David I. Ketcheson Lajos Lóczi Adrián Németh March 21,
More informationPOSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS. Cheng Wang. Jian-Guo Liu
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS SERIES B Volume 3 Number May003 pp. 0 8 POSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS
More informationSpectral collocation and waveform relaxation methods with Gegenbauer reconstruction for nonlinear conservation laws
Spectral collocation and waveform relaxation methods with Gegenbauer reconstruction for nonlinear conservation laws Z. Jackiewicz and B. Zubik Kowal November 2, 2004 Abstract. We investigate Chebyshev
More informationApplication of the Kurganov Levy semi-discrete numerical scheme to hyperbolic problems with nonlinear source terms
Future Generation Computer Systems () 65 7 Application of the Kurganov Levy semi-discrete numerical scheme to hyperbolic problems with nonlinear source terms R. Naidoo a,b, S. Baboolal b, a Department
More information