Numerical Finite Differences Solution Convergence 1. Does the discrete solution i retain the qualitative propeties of the continuous solution u(a)? 2. Does the solution become more accurate when △x-0? 3 Can we make a()-jfor0≤j≤m+1 arbitrarily small? SMA-HPC⊙2003MT Finite Differences 10
Discretization Properties of A-I Error Analysis Let A-={ai}1≤i≤m ● Non-negativity N11 c≥0,for1≤i,≤m ● Boundedness N12 0≤∑a for 1<i<nm SMA-HPC⊙2003MT Finite Differences 11
Discretization Qualitative Properties of i Error Analysis f≥0→}t≥0 Af f;=f(c)≥0,for1≤j≤n Then a;=∑a3≥0,tor1≤a≤m SMA-HPC⊙2003MT Finite Differences 12
Discretization Qualitative Properties of i Error Analysis Discrete Stability i=A f |=maxa=max(∑af) ≤max(>a) max fil SMA-HPC⊙2003MT Finite Differences 13
Discretization Truncation Error Error Analysis For any v EC4 we can show that N13 0(a+)-20(a)+0(21)=0"(a)+-1((+0△m) △ △a2 1<6<1 Take≡t(-u"=f u(c1+1)-2u(a1)+(c1-1) f( )-70u4(a1+0△a) 12 SMA-HPC⊙2003MT Finite Differences 14