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 U<0 0 1 a(a, t) u(a-Ut), if a-Ut<1 91(t-U), if a-Ut>1 SMA-HPC⊙2003MT Hyperbolic Equations 4