麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 12 Numerical Schemes for Scalar

Numerical Schemes for Scalar One-Dimensional Conservation Laws Lecture 12

Finite Volume Computational Cells Discretization t=T 7+1 c=△ t=m△t 011-1jj+-1J U SMA-HPC⊙2003MT Hyperbolic Equations 1

Finite Volume Cell averages Discretization We think of in as representing cell averages 1 a(a, t") da C SMA-HPC⊙2003MT Hyperbolic Equations 2

Conservative Definition Methods Applying integral form of conservation law to a cell 3 d dt ad=-f((+,1)-f(a(a=,D) suggests in+l △t△a= T+1= △t C j+号 F SMA-HPC⊙2003MT Hyperbolic Equations 3

Conservative Numerical Flux function Methods +=F(a1b,①31+1,…,,…,句+) and F is a numerical flux function of l+r+1 arguments that satisfies the following consistency condition F(u, a 儿。L )=f(u) 7+1 SMA-HPC⊙2003MT Hyperbolic Equations 4

Conservative Lax-Wendroff Theorem Methods If the solution of a conservative numerical scheme converges as A-,0 with At fixed, then it converges to a weak solution of the conservation law shock capturing schemes are possible N2 SMA-HPC⊙2003MT Hyperbolic Equations 5

Conservative Lax-Wendroff Theorem Methods Shock Capturing In the exact problem da=-(fo-fJ) dt co A conservative numerical scheme satisfies an analogous discrete condition N3 △c J 4t2(2 7+1 )=一∑(F+-F-3) 0 J+ SMA-HPC⊙2003MT Hyperbolic Equations 6

Conservative First Order Upwind Methods Linear Advection Equation +a 0 a constant> 0 at aa 7+1 △t UP UP Let FUP= ai UP -=a-1 +1 △ta (a SMA-HPC⊙2003MT Hyperbolic Equations 7

Conservative First Order Upwind Methods Linear Advection Equation What about a 0 1 a<0 or 7+1 a△t a△t 20 2△c 2△c +1 SMA-HPC⊙2003MT Hyperbolic Equations 8

Conservative First Order Upwind Methods Linear Advection Equation In conservative form +1 △t LFl +-P UPn △c 1 +=20(1+1+a) UP a(y+1- F UP au a>0 UP j auj+l a<0 SMA-HPC⊙2003MT Hyperbolic Equations 9