The Finite Difference Scheme Of Wave Equation With Special Boundary Conditions

The stabilization control of wave equation is an important research con-tent for distributed parameter control theory. The control equation is usually wave equation with initial and boundary (IB V) problem with feedback bound-ary conditions. One of them is wave equation with Neumann damped bound-ary. There is important theoretical significance and application value for us to study its numerical algorithm.Firstly, a three-level implicit finite difference scheme is constructed for the IBV problem of one-dimension wave equation with a special boundary condition,wtt(x,t) - wxx(x, t) = f(x, t), (x,t) ? (0,1) × (0,T],wx(0,t) - w(0,t) = 0, wx(1,t)=-wt(1,t), = t ? [0,T], (3)w(x,0) = ?(x), wt(x, 0) = ?(x), X ? [0,1],where Robin boundary is on the left and Neumann damped boundary is on the right. The existence and uniqueness are proved. The discretized energy method is employed to show that the difference scheme is two-order conver-gent in maximum norm with respect to time and space directions and uncon-ditional stable with respect to initial conditions and right-hand side term. The numerical experiment verifies the theoretical results.Next, wave equation IBV problem is changed to the following equiva-lent weak coupled hyperbolic equations in the manner of introducing a novel variable.wt(x, t) + wx(x, t) = v(x,t), (x, t) G (0,1) × (0, T],vt(x,t) - vx(x,t) = f(x,t), (x,t) ? (0,1) × (0,T], (4)w(x, 0) = ?(x), v(x, 0)= ?(x) + ?'(x), x ? [0,1],w(0,t) = v(0,t) - v(1,t) = 0, t ? [0,T].Then a novel finite difference scheme of wave equation IBV problem is con-structed (3), which are derived by the finite difference scheme of coupled hyperbolic equations (4). The discrete energy method is employed to show that the difference scheme is two-order convergent in both time and space and unconditionally stable with respect to both initial conditions and right-hand term in L2 norm. A higher order convengent difference scheme is obtained by Richardson extrapolation. The accuracy and efficiency of the scheme are checked by numerical experiments.Finally, a compact finite difference scheme is construced for above wave equation IBV problem, and it is proved that it is two-order convergent about time and four-order convergent about space in L? norm by numerical experi-ment.