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