(六B)双曲型守恒律及可压缩流的高分辨率格式 - LSEC.PPT

(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) 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: Minmod Flux Limiter on Sweby Diagram Superbee Flux Limiter on Sweby Diagram MC Flux Limiter on Sweby Diagram van Leer Flux Limiter Summary of Some Flux Limiting Functions 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 ： 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)) 权重 r=3(ENO) 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) * 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 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: monotone schemes and TVD and TVD schemes are monotonicity preserving schemes 1 2 3 1 2 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). 1 2 3 1 2 The Superbee limiter applies the minimum limiting and maximum steepeningpossible to remain TVD. It i