Linear Fourth-order Problems Research Articles

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.

Application of the fourth-order Runge-Kutta method for the solution of high-order general initial value problems

A Runge-Kutta algorithm for nth order initial value problem (IVP), x ( n) ( t) = f( t, x, x′,…, x ( n−1) , t > t 0, with x( t 0), x ( r) ( t 0) given ( r = 1,2,…, n −1), is developed by means of the algorithm used for the first-order IVP. As an example, the method is tested on a fourth-order IVP involving nonlinear terms and on a linear fourth-order problem from ship dynamics.

A family of numerical methods for the solution of high-order general initial value problems

A family of numerical methods is developed for the solution of the general initial value problem y ( N) ( t) = f( t, y, y t , …, y ( N−1) ), t> t 0, with y( t 0, y ( r) ( t 0) given ( r = 1, 2, …, N−1). The orders of the methods are seen to be one or two and global extrapolation is used to increase the order of a given method by one or two powers. The methods are tested on the Blasius nonlinear third-order initial value problem and on a linear fourth-order problem from ship dynamics.