A two-level implicit difference scheme using three spatial grid points of Crandall form of O(k2 + kh2 + h4) is obtained for solving the one-dimensional quasilinear parabolic partial differential equation, u, = f ( x , t, u, u , , u,) with Dirichlet boundary conditions. The method, when applied to a linear convection-diffusion problem, is shown to be unconditionally stable. The numerical results show that the proposed method produces accurate and oscillation-free solutions.

We present the fourth-order finite difference methods for the system of 2D nonlinear elliptic equations using 9-grid points on a square region R subject to Dirichlet boundary conditions. The method has been tested on viscous, incompressible 2D Navier-Stokes equations. The numerical results show that the proposed methods produce accurate and oscillation-free solutions for large Reynolds numbers.

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We present a nine-point fourth-order finite difference method for the nonlinear secondorder elliptic differential equation .4u, + Bu,, = f ( x , y , u , u,, u,) on a rectangular region R subject to Dirichlet boundary conditions u ( x , y ) = g ( x , y ) on aR. We establish, under appropriate conditions O(h4)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.

A C1‐continuous time‐domain spectral finite element (SFE) is developed for efficient and accurate analysis of flexural‐guided wave propagation in Euler–Bernoulli beam‐type structures. A new C1‐continuous spectral interpolation using the Lobatto basis is proposed, which is shown to eliminate the Runge phenomenon observed in the conventional higher order Hermite interpolation. It is also able to diagonalize the mass matrix, an attractive feature of existing C0‐continuous SFEs, which enhances computational efficiency. The developed element is validated by comparing the results for natural frequencies of first 20 modes with analytical solutions, and its performance for wave propagation problems is assessed in comparison with converged ABAQUS solutions obtained with a very fine mesh using the classical beam element. It is shown that the present element yields excellent accuracy with much faster convergence, higher computational efficiency, and many‐fold reduction in computational time than the conventional FE for narrowband high‐frequency flexural guided wave propagation problems in both undamaged and damaged beams. It also shows excellent performance for wave propagation under broadband impact excitations and initial displacements. The C1‐continuous interpolation proposed here will pave the way for developing several new SFEs for elastic‐ and piezoelectric‐laminated beams using advanced higher order laminated theories, which require C1‐continuity of displacements.