(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, DG… 6B.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

6B.1 TVD格式(续) 带通量限制子的二阶TVD

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 each of its arguments Example: 1st order upwind scheme

is said to be Total Variation Diminishing (TVD) if Lecture 6 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) That is: monotone schemes and TVD and TVD schemes are monotonicity preserving schemes

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: 2 1 1 2 3

Minmod Flux Limiter on Sweby Diagram 2 1 1 2 3 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).

Superbee Flux Limiter on Sweby Diagram 2 1 1 2 3 The Superbee limiter applies the minimum limiting and maximum steepening possible to remain TVD. It is known to suffer from excessive sharpening of slopes 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 steepening of the slopes.

MC Flux Limiter on Sweby Diagram 2 1 1 2 3 The MC limiter transitions from upwind (theta<0) to Fromm (at theta=1/3) then switches to a constant(at theta=3). This is a compromise between Superbee and minmod

van Leer Flux Limiter The van Leer limiter charts a careful compromise path through the Sweby TVD region.

Summary of Some Flux Limiting Functions Linear non-TVD limiters Nonlinear second order TVD limiters

6B.5 WENO格式

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。

ENO的候选模板（上风情况）

迎风型WENO格式

数值通量： 低阶ENO的线性组合得到高阶的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 ：

注：f是cell average

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.

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))

q中的f为cell average

权重

r=3(ENO)

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。

负特征值的通量

推广到双曲型方程组情况

Steger-Warming: characteristic-wise

L-F: component-wise

Jiang-Wu implementation Jiang GS, Wu CC, JCP 150, 561-591, 1999

Jiang-Wu implementation (2)

Jiang-Wu implementation (3) System of conservation laws

Jiang-Wu implementation (4)