Advanced Computational Fluid Dynamics AA215A Lecture 5 Antony Jameson Winter Quarter 2012 Stanford CA Abstract Lecture 5 shock capturing schemes for scalar conservation laws Contents 1 Shock Capturing Schemes for Scalar Conservation Laws 1 1 Introduction 1 2 The need for oscillation control 1 2 1 Odd even de coupling 1 2 2 Propagation of a step discontinuity 1 3 Iserles barrier theorem 1 4 Stability in the L norm 1 5 Local extremum diminishing LED schemes 1 6 Total variation diminishing TVD schemes 1 7 Semi discrete L stable and LED schemes 1 8 Growth of the L norm with a source term 1 9 Accuracy limitation on L stable and LED schemes 1 10 Artificial diffusion and upwinding 1 11 Artificial Diffusion and LED Schemes for Nonlinear Conservation Laws 1 12 The First Order Upwind Scheme 1 13 Shock Structure of the Upwind Scheme 1 14 Upwinding and Conservation 1 15 The Engquist Osher Upwind Scheme 1 16 The Jameson Schmidt Turkel Scheme 1 17 Essentially Local Extremum Diminishing ELED Schemes 1 18 Symmetric Limited Positive SLIP Schemes 1 19 Upstream Limited Positive USLIP Schemes 1 20 Reconstruction 1 21 Reconstruction for a Nonlinear Conservation Law 1 22 SLIP Reconstruction 1 2 3 3 3 4 5 5 7 8 10 11 12 14 15 16 17 18 19 21 23 23 28 28 30 33 Lecture 5 Shock Capturing Schemes for Scalar Conservation Laws 1 1 Introduction A major achievement in the early development of computational fluid dynamics CFD was the formulation of non oscillatory shock capturing schemes The first such scheme was introduced by Godunov in his pioneering work first published in 1959 Godunov also showed that nonoscillatory schemes with a fixed form are limited to first order accuracy This is not sufficient for adequate engineering simulations Consequently there were widespread efforts to develop high resolution schemes which circumvented Godunov s theorem by blending a second or higher order accurate scheme in smooth regions of the flow with a first order accurate non oscillatory scheme in the neighborhood of discontinuities This is typically accomplished by the introduction of logic which detects local extrema and limits their formation or growth Notable early examples include Boris and Books flux corrected transport FCT scheme published in 1973 and Van Leer s Monotone Upstream Conservative Limited MUSCL scheme published in 1974 Here we first discuss the formulation of non oscillatory schemes for scalar conservation laws in one or more space dimension and illustrate the construction of schemes which yield second order accuracy in the bulk of the flow but are locally limited to first order accuracy at extrema Next we discuss the formulation of finite volume schemes for systems of equations such as the Euler equations of gas dynamics and analyze the construction of interface flux formulas with favorable properties such as sharp resolution of discontinuities and assurance of positivity of the pressure and density The combination of these two ingredients leads to a variety of schemes which have proved successful in practice 1 2 1 2 1 The need for oscillation control Odd even de coupling Consider the linear advection equaton for a right running wave u u a 0 a 0 1 1 t x Representing the discrete solution at meshpoint by vj a semi discrete scheme with central differences is dvj a vj 1 vj 1 0 1 2 dt 2 x 3 CHAPTER 1 SHOCK CAPTURING SCHEMES FOR SCALAR CONSERVATION LAWS 4 Figure 1 1 Odd even mode j 1 j j 1 Figure 1 2 Propagation of right running wave Then an odd even mode vj 1 j 1 3 dvj 0 dt 1 4 gives Thus an odd even mode is a stationary solution and odd even decoupling should be removed via the addition of artificial diffusion or upwinding 1 2 2 Propagation of a step discontinuity Consider the propagation of a step as a right running wave by the central difference scheme Now the discrete derivative vj 1 vj 1 Dx vj 0 1 5 2 x and hence with a 0 dvj 0 1 6 dt giving an overshoot On the other hand the upwind scheme Dx vj vj vj 1 x 1 7 correctly yields dvj 0 dt 1 8 CHAPTER 1 SHOCK CAPTURING SCHEMES FOR SCALAR CONSERVATION LAWS 5 wave propagation r points s points Figure 1 3 Barrier theorem 1 3 Iserles barrier theorem Both the need to suppress odd even modes and the need to prevent overshoots in the propagation of a step discontinuity motivate the use of an upwind scheme However purely upwind schemes are also subject to limitations as a consequence of Iserles barrier theorem Consider the approximation of the linear advection equation by a semi discrete scheme with r upwind points and s downwind points The theorem states that the maximum order of accuracy of a stable scheme is min r s 2r 2s 2 1 9 This is a generalization of an earlier result of Engquist and Osher that the maximum order of a accuracy of a stable upwind semi discrete scheme is two s 0 in formula 1 9 It may also be compared to Dahquist s result that A stable linear multistep schemes for ODEs are at most second order accurate One may conclude from Iserles theorem that upwind biased schemes may be preferred over purely upwind schemes as a route to attaining higher order accuracy 1 4 Stability in the L norm Consider the nonlinear conservation law for one dependent variable with diffusion u u f u u 0 t x x x 1 10 With zero diffusion 1 10 is equivalent in smooth regions to u u a u 0 t x 1 11 where the wave speed is a u f u The solution is constant along characteristics x a u t so extrema remain unchanged as they propagate unless the characteristics converge to form a shock wave a process that does not increase extrema With a positive diffusion coefficient the right hand side of equation 1 10 is negative at a maximum and positive at a minimum Thus in a true solution of 1 10 extrema do not increase in absolute value It follows that L stability is an CHAPTER 1 SHOCK CAPTURING SCHEMES FOR SCALAR CONSERVATION LAWS 6 appropriate criterion for discrete schemes consistent with the properties of true solutions of the nonlinear conservation law 1 10 Consider the general discrete scheme aij vjn 1 12 vin 1 j where the solution at time level n 1 depends on the solution over an arbitrarily large stencil of points at time level n A Taylor series expansion of 1 12 about the point xi at time t yields t2 2 vj vj xi xi t 2 t2 v xi xj xi 2 2 v xi aij v xi xj xi x 2 x2 v xi t 1 13 j For consistency with any equation with no source term as is the case for equation 1 10 aij 1 1 14 j Also it follows from 1 12 that vin 1 j aij vjn and hence v n 1 max i j j aij v n aij v n …