On the convergence of a three-layer semi-discrete scheme for the nonlinear dynamic Kirchhoff string equation

Abstract

Abstract In this work, the initial-boundary value problem is considered for the dynamic Kirchhoff string equation u t ⁢ t - ( α ⁢ ( t ) + β ⁢ ∫ - 1 1 u x 2 ⁢ d x ) ⁢ u x ⁢ x = f u_{tt}-\bigl{(}\alpha(t)+\beta\int_{-1}^{1}u_{x}^{2}\,\mathrm{d}x\bigr{)}u_{xx}=f . Here α ⁢ ( t ) \alpha(t) is a continuously differentiable function, α ⁢ ( t ) ≥ c 0 > 0 \alpha(t)\geq\mathrm{c}_{0}>0 and 𝛽 is a positive constant. For solving this problem approximately, a symmetric three-layer semi-discrete scheme with respect to the temporal variable is applied, in which the value of a nonlinear term is taken at the middle point. This approach allows us to find numerical solutions per temporal steps by inverting the linear operators. In other words, applying this scheme, a system of linear ordinary differential equations is obtained. The local convergence of the scheme is proved. The results of numerical computations using this scheme for different test problems are given for which the Legendre–Galerkin spectral approximation is applied with respect to the spatial variable.