麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 4 Finite Difference Discretization

Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Problems Lecture 4

Finite Difference Problem Definition Formulas Given l+r+ 1 distinct points (2-1,3-+1,……,0,…,r), find the weights o such that dmy S!, dam IS of optimal order of accuracy N1 TWo approaches Lagrange interpolation o Undetermined coefficients SMA-HPC⊙2003MT Finite Differences 1

Finite Difference Lagrange interpolation Formulas Lagrange polynomials c- - Li(a) 1)(m-+1)…( - 1)…(x;-21-1)(c-m13+1)…(a Lagrange interpolant 6(a)=∑L(a) SMA-HPC⊙2003MT Finite Differences 2

Finite Difference Lagrange interpolation Formulas Approximate do d dL da =:0 doc lc=co j=-I doc x=ro Therefore dmL Te drm 0 SMA-HPC⊙2003MT Finite Differences 3

Finite Difference Lagrange interpolation Formulas Example sett=7=1,(a-1,j,m+1 Second order Lagrange interpolant 0(c) (m-21)(m-2+1) a 0-1+ )(x-31+1) (2-1j-1)(mj-mj+1 0+ 1-0j+1 (m-1-1)(m-y) +1 SMA-HPC⊙2003MT Finite Differences 4

Finite Difference Lagrange interpolation Formulas Example Assuming a uniform grid m=1(First derivative) 616 +1 2△x△a △ Forward -2△0 2△ Centered +1 3 2△m△a2△m Backward SMA-HPC⊙2003MT Finite Differences 5

Finite Difference Lagrange interpolation Formulas Example m=2( Second derivative) y+1 2 Centered N2 SMA-HPC⊙2003MT Finite Differences 6

Finite Difference Undetermined coefficients Formulas Start from dmu dam Insert Taylor expansions for v; about a = i 00+0(x;-x)+2v(x1;-)2+…, determine coefficients 8, to maximize accuracy SMA-HPC⊙2003MT Finite Differences 7

Finite Difference Undetermined coefficients Formulas Example m=2, l=r=1, i=0,(uniform spacing Ac) 0=621(0-△a+20-40+0+…) +6v0 +62(0+△m+2n+0"+a(0)+…) SMA-HPC⊙2003MT Finite Differences 8

Finite Difference Undetermined coefficients Formulas Example Equating coefficients of u() k=0→021+6+62=0 k=1→△a(62-621)=0 k=2→22(61+62)=1 Solve 163=-△m2 2 1 3N4 SMA-HPC⊙2003MT Finite Differences 9