1、1,(六B)双曲型守恒律及可压缩流的高分辨率格式,TVD, NND, MUSCL,ENO,WENO, 群速度控制法，Godunov, Roe, CESE, DG6B.1 TVD格式(续)6B.2 MUSCL格式6B.3 NND格式6B.4 群速度控制法6B.5 WENO格式,2,6B.1 TVD格式(续),带通量限制子的二阶TVD,3,单调格式(monotone scheme),A scheme,with non-negative integers, is said to be monotone if,That is, H is a non-decreasing function of eac
2、h of its arguments,Example: 1st order upwind scheme,4,TVD 格式的定义,is said to be Total Variation Diminishing (TVD) if,单调保持格式的定义 Monotonicity Preserving (MP) Scheme. A scheme is said to be MP if whenever the data is monotone the solution is monotone in the same sense.,Theorem (Harten),Lecture 6,That is:
3、 monotone schemes and TVD and TVD schemes are monotonicity preserving schemes,5,带通量限制子的TVD,To guarantee 2nd order accuracy and avoid excessive compression of solutions, Sweby suggested the following TVD region as a suitable range for the flux limiting function:,1,2,3,1,2,6,Minmod Flux Limiter on Swe
4、by Diagram,1,2,3,1,2,It is apparent that the minmod flux limiter applies the maximum possible limiting allowed within the second order TVD region.(i.e. it will be rather dissipative and smear out discontinuities somewhat as seen on the right hand side figure).,7,Superbee Flux Limiter on Sweby Diagra
5、m,1,2,3,1,2,The Superbee limiter applies the minimum limiting and maximum steepeningpossible to remain TVD. It is known to suffer from excessive sharpening ofslopes as a result. On the right we show what happens to a smooth sine wave after 20 periods.Notice the flattening of the peaks and the steepe
6、ning of the slopes.,8,MC Flux Limiter on Sweby Diagram,1,2,3,1,2,The MC limiter transitions from upwind (theta0) to Fromm (at theta=1/3) then switches to a constant(at theta=3). This is a compromise between Superbee and minmod,9,van Leer Flux Limiter,The van Leer limiter charts a careful compromise
7、path throughthe Sweby TVD region.,10,Summary of Some Flux Limiting Functions,Nonlinear second orderTVD limiters,Linear non-TVD limiters,11,6B.5 WENO格式,12,WENO的基本思想,设所有网格点的Cell average 已知 ，要求对变量h的逼近函数p满足：WENO的思想是将ENO的三个候选模板的逼近进行加权，使所得的逼近函数满足上述要求。,13,ENO的候选模板（上风情况）,14,迎风型WENO格式,15,数值通量：低阶ENO的线性组合得到高阶的
8、WENO通量，组合系数称为权重。考虑迎风效应，r-ENO的候选模板有r个 ：,16,注：f是cell average,17,适用于包含间断的模板，但对于光滑模板，浪费了其他r-1模板.,18,r个模板合在一起有2r-1个点( )，按照(4)式可给出网格界面处2r-1阶逼近(对原始函数f(x),19,20,q中的f为cell average,21,权重,22,r=3(ENO),23,精度,未必有错，最终是导数的逼近精度而不是通量的逼近精度。,24,负特征值的通量,25,推广到双曲型方程组情况,26,Steger-Warming: characteristic-wise,27,L-F: component-wise,28,Jiang-Wu implementation,Jiang GS, Wu CC, JCP 150, 561-591, 1999,29,Jiang-Wu implementation (2),30,Jiang-Wu implementation (3),System of conservation laws,31,Jiang-Wu implementation (4),
