麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 8 Finite Difference Discretization of Hyperbolic Equations

Finite Difference Discretization of Hyperbolic Equations. Linear Problems Lectures 8.9 and 10

First Order Wave Equation INITIAL BOUNDARY VALUE PROBLEM(IBVP du +U at da 0,∈(0,1) Initial condition: u(x,0)=u(a) Boundary conditions:u(o, t)=go(t)if U>0 u(1,t)=g1(t)iU<0 SMA-HPC⊙2003MT Hyperbolic Equations 1

First Order Wave Solution Equation du du d du d at dt+odac aa ot dt dx dm dtU a =Ut+E Characteristics du=0,=u(a, t)=f(s)=f(e-Ut) General solution SMA-HPC⊙2003MT Hyperbolic Equations 2

First Order Wave Solution Equation U>0 0 a(a, t) u (a-Ut), if c-Ut>0 go(t-U), if a-Ut<o SMA-HPC⊙2003MT Hyperbolic Equations 3

First Order Wave Solution Equation U1 SMA-HPC⊙2003MT Hyperbolic Equations 4

First Order Wave Stability Equation L2([0,1])norm lul2(t) a2(ac, t)da 0 ot tuou au d=0 atl=-U(x2(1,)-(0,) SMA-HPC⊙2003MT Hyperbolic Equations 5

Model Problem +U t 0—0 0,∈(0,1) Initial condition u(x,0)=0(x) Periodic Boundary conditions: u(0, t)=u(1, t) d u|2=0→‖u|2(t)=|u0|2= constant SMA-HPC⊙2003MT Hyperbolic Equations 6

Example Model Problem Periodic Solution(U >0) t=0 05 t= 0.1 0.2 t=2 6 0.8 X SMA-HPC⊙2003MT Hyperbolic Equations 7

Finite Difference Discretization Solution Discretize(0, 1)into J equal intervals Ac 1 △c c5=△ and(0, T) into N equal intervals At T △t N tn=n△t 0<<J a≈w=(,t2),for 0<m<N SMA-HPC⊙2003MT Hyperbolic Equations 8

Finite Difference Discretization Solution a=1 t=T +1+… △t 1+… 1j-1j+1J SMA-HPC⊙2003MT Hyperbolic Equations 9