Difference schemes for fully nonlinear pseudo-hyperbolic systems

The general difference schemes for the first boundary problem of the fully nonlinear pseudohyperbolic systems $$f(x,t,u,u_x ,u_{xx} ,u_t ,u_{tt} ,u_{xt} ,u_{xxt} ) = 0$$ are considered in the rectangular domainQ _T ={0≤x≤l, 0≤t≤T}, whereu(x, t) andf(x, t, u, p ₁,p ₂,r ₁,r ₂,q ₁,q ₂) are twom-dimensional vector functions withm ≥ 1 for (x, t) ∈Q _T andu, p ₁,p ₂,r ₁,r ₂,q ₁,q ₂ ∈R ^m . The existence and the estimates of solutions for the finite difference system are established by the fixed point technique. The absolute and relative stability and convergence of difference schemes are justified by means of a series of a prior estimates. In the present study, the existence of unique smooth solution of the original problem is assumed. The similar results for nonlinear and quasilinear pseudo-hyperbolic systems are also obtained.