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? We can write m2+1 a△t 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