Finite Difference First Order Upwind Scheme Solution Matrix Form We can write i= si 1 n A0 0 (1-C)0 (1-C) 00 0 000 0 0C(1-C) SMA-HPC⊙2003MT Hyperbolic Equations 15
Finite Difference First Order Upwind Scheme Solution Example t+ 0 EXACT t=0.5 t=0.75 100 C- At =0.5 T=1→N=200% 10.20304050.6070. X SMA-HPC⊙2003MT Hyperbolic Equations 16
Definition Convergence The finite difference algorithm converges if m a-x2l=0,1≤m≤N △a,△t→0 N△t=T J△w=1 for any initial condition u (ac) c=△c∑03)=△a|l2N2 SMA-HPC⊙2003MT Hyperbolic Equations 17
Definition Consistency The difference scheme li=0 is consistent with the differential equation Lu=0 f For all smooth functions v (C0)-(C)y→0,for ≤J <n<N When△,△t-0 SMA-HPC⊙2003MT Hyperbolic Equations 18
First Order Upwind Scheme Consistency Difference operator 1 0 +1 △t Differential operator 60 C +U ot d SMA-HPC⊙2003MT Hyperbolic Equations 19