On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system (Q2804375)

The subject of this paper is the numerical approximation of the so-called Klein-Gordon-Zakharov system of partial differential equations (KGZ) which is of the following form: NEWLINE\[NEWLINE \partial_{tt}\psi(x,t)-\Delta\psi(x,t)+\psi(x,t)+\psi(x,t)\phi(x,t) +\lambda|\psi|^2\psi(x,t)=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\partial_{tt}\phi(x,t)-\Delta\phi(x,t)-\Delta(|\psi(x,t)|^2)=0,NEWLINE\]NEWLINE with given initial conditions for \(\psi(x,0)\), \(\partial_t\psi(x,0)\), \(\phi(x,0)\), and \(\partial_t\phi(x,t)\). Here the one-dimensional space approximation of the KGZ system on a finite interval \((a,b)\) is considered with additional zero Dirichlet boundary conditions at \(a\) and \(b\). A remark concerning Neumann conditions is given.NEWLINENEWLINEFor the space sine pseudospectral method the standard \(L^2\) projection \(P_N:L^2(a,b)\rightarrow X_N\) and the trigonometric interpolation operator \(I_N:C(a,b)\rightarrow X_N\) are used, where NEWLINE\[NEWLINEX_N=\mathrm{span} \{\phi_l=\sin(\mu_l(x-a)):x\in [a,b],\,\mu_l={{\pi l}\over(b-a)},l=1,2,\dots,N-1\}NEWLINE\]NEWLINE (orthogonality of the basic functions of interpolation is important!). The projection operators need integration, hence for approximation the numerical quadrature has to be applied (ex. trapezoidal rule). Finally, an elegant explicit algorithm is obtained, which turns out to be time-reversible (i.e., it works with positive and negative time step.) The author presents error bounds for sufficiently smooth solutions, and deduces the (conditional) stability-convergence result. Some numerical examples are given and a long-time energy conservation test is joint.