A nonlinear shooting method and its application to nonlinear Rayleigh-Bénard convection (Q372481)

Summary: The simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type \(\mathbf{X}'(z) = f(z, \text\textbf{X}(z))\), \(\mathbf{X}(z) \in \mathbb R^n\), \(n > 1\), \(z \in [0, 1]\) is considered where \(m\) components of \(\mathbf{X}(z)\) are known at one of the boundaries and \((n - m)\) components of \(\mathbf{X}(z)\) are specified at the other boundary. The map \(f\) is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of \textit{S. N. Ha} [Comput. Math. Appl. 42, No. 10-11, 1411--1420 (2001; Zbl 0999.65077)] to the BVPs arising in the higher-order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-Bénard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role.