Finite Difference Discretization Solution NOTATION 0r approximation to v(a,, tm)=vj UER vector of approximate values at time n; 2={07 unEIR vector of exact values at time n n2={v(x,t)}=1 SMA-HPC⊙2003MT Hyperbolic Equations 10
Finite Difference Approximation Solution For example..(forU> O 00 a(a,, tn) 0(-1 0 0(c3,t2+1)-v(x,t) 02+1 t △t △t Forward in Time Backward (Upwind) in Space SMA-HPC⊙2003MT Hyperbolic Equations 11
Finite Difference First Order Upwind Scheme Solution wt +Uur=0 suggests △t+- am+1- 0→ △a 1<1<J a+1=-C(a-0-1) 0<m<N A=iT 0<m<N Courant number C=U△t/△a SMA-HPC⊙2003MT Hyperbolic Equations 12
Finite Difference First Order Upwind Scheme Solution Interpretation +1 dx U P +1 △t}:C△m Use Linear Interpolation -1,j -2j-13+1 uP≈Ca1+(1-C) SMA-HPC⊙2003MT Hyperbolic Equations 13
Finite Difference First Order Upwind Scheme Solution Explicit Solution U>0。 known values T x unknown values XXXXXXX ●Xxx no matrix inversion i exists and is unique 1 1 C(-a-1) SMA-HPC⊙2003MT Hyperbolic Equations 14