Solvability of concatenated Runge–Kutta equations for second-order nonlinear PDEs

In this paper, we consider the solvability of concatenated Runge–Kutta (CRK) equations, which are the nonlinear algebraic equations arising from CRK methods applied to some kind of second-order nonlinear partial differential equations. To guarantee the existence and uniqueness of the solution of CRK equations, temporal and spatial step-sizes of CRK methods should be chosen appropriately. Under different assumptions, with the homotopy continuation method, we choose different step-sizes to ensure the solvability of CRK equations.