Advanced Computational Fluid DynamicsAA215A Lecture 5Antony JamesonWinter Quarter, 2012, Stanford, CAAbstractLecture 5 shock capturing schemes for scalar conservation lawsContents1 Shock Capturing Schemes for Scalar Conservation Laws 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The need for oscillation control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Odd-even de-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Propagation of a step discontinuity . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Iserles’ barrier theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Stability in the L∞norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Local extremum diminishing (LED) schemes . . . . . . . . . . . . . . . . . . . . . . . 71.6 Total variation diminishing (TVD) schemes . . . . . . . . . . . . . . . . . . . . . . . 81.7 Semi-discrete L∞stable and LED schemes . . . . . . . . . . . . . . . . . . . . . . . . 101.8 Growth of the L∞norm with a source term . . . . . . . . . . . . . . . . . . . . . . . 111.9 Accuracy limitation on L∞stable and LED schemes . . . . . . . . . . . . . . . . . . 121.10 Artificial diffusion and upwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.11 Artificial Diffusion and L E D S chemes for Nonlinear Conservation Laws . . . . . . . . 151.12 The First Order Upwind Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.13 Shock Structur e of the Upwind Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 171.14 Upwinding and Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.15 The Engquist-Osher Upwind Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.16 The Jameson-Schmidt-Turkel Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 211.17 Essentially Local Extremum Diminishing (ELED) Schemes . . . . . . . . . . . . . . 231.18 Symmetric Limited Positive (SLIP) Schemes . . . . . . . . . . . . . . . . . . . . . . 231.19 Upstream Limited Positive (USLIP) Schemes . . . . . . . . . . . . . . . . . . . . . . 281.20 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.21 Reconstruction for a Nonlinear Conservation Law . . . . . . . . . . . . . . . . . . . . 301.22 SLIP Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Lecture 5Shock Capturing Schemes for Scala rConservation Laws1.1 IntroductionA major achievement in the early development of computational fluid dynamics (CFD) was theformulation of non-oscillatory shock capturing schemes. The first su ch scheme was introducedby Godunov in his pioneering work first published in 1959 (?). Godunov also showed that non-oscillatory schemes with a fixed form are limited to first order accuracy. This is not sufficient foradequate engineering simulations. Consequently there were widespread efforts to develop “highresolution” schemes which circumvented Godunov’s theorem by blend ing a second or higher orderaccurate scheme in smooth regions of the flow with a first order accurate non-oscillatory schemein the neighborhoo d of discontinuities. This is typically accomplished by the introd uction of logicwhich detects local extrema and limits their formation or growth. Notable early examples includeBoris and Books’ flux corrected transport (FCT) scheme published in 1973 (?) and Van Leer’sMonotone Upstream Conservative Limited (MUSCL) scheme published in 1974 (?).Here we first discuss the formulation of non-oscillatory schemes for scalar conservation laws inone or more space dimension, and illustrate the construction of schemes which yield second ord eraccuracy in the bulk of the flow bu t are locally limited to first order accuracy at extrema. Nextwe discuss the formulation of finite volume schemes for systems of equations such as the Eulerequations of gas dynamics, and analyze the construction of interface flux formulas with favorableproperties such as sharp resolution of discontinuities and assurance of positivity of the pressure anddensity. The combination of these two ingredients leads to a variety of schemes which have provedsuccessful in practice.1.2 The need for oscillation control1.2.1 Odd-even de-couplingConsider the linear advection equaton for a r ight running wave∂u∂t+ a∂u∂x=0, a > 0. (1.1)Representing the discrete solution at meshpoint by vj, a semi-discrete scheme with central differ-ences isdvjdt+a2∆x(vj+1− vj−1) = 0 (1.2)3CHAPTER 1. SHOCK CAPTURING SCHEMES FOR SCALAR CONSERVATION LAWS 4Figure 1.1: Odd -even modej−1 j+1jFigure 1.2: Propagation of right running waveThen an odd -even modevj=(−1)j(1.3)givesdvjdt=0. (1.4)Thus an odd-even mode is a stationary solution, and o dd-even decoupling should be removedvia the addition of artificial diffusion or upwinding.1.2.2 Propagation of a step discontinuityConsider the propagation of a step as a right running wave by the central differen ce scheme. Nowthe discrete derivativeDxvj=vj+1− vj−12∆x< 0 (1.5)and h ence with a>0dvjdt> 0 (1.6)giving an overshoot. On the other hand the upwind schemeD−xvj=vj− vj−1∆x(1.7)correctly yieldsdvjdt=0. (1.8)CHAPTER 1. SHOCK CAPTURING SCHEMES FOR SCALAR CONSERVATION LAWS 5r points s pointswave propagationFigure 1.3: Barrier …