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A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity.

Compact difference methods, essentially the implicit versions of finite different methods, are superior to the explicit versions in achieving high order accuracy. High order compact methods directly approximate all derivatives with high accuracy without any variational operation or projection. They are very effective for complex and strongly nonlinear partial differential equations, especially when variational methods are inconvenient to implement. They have smaller stencils than explicit finite difference methods while maintaining high order accuracy.

Much development has been achieved in finite difference methods. A large number of papers related to various finite difference algorithms have been published over the past decades. Kreiss [

Compact methods are widely used in applications in engineering and physics. Ekaterinaris [

Irregular geometries could hinder the application of compact methods. To facilitate nonuniform discretization, one way is to use a transformation so that compact methods can be implemented in the transformed space where the nominal rate of accuracy could be achieved. For example, zeros of Chebyshev polynomials could be adopted as the grid points for computation [

In the past two decades, Boussinesq-type equations, derived from depth-integration of conservation laws for mass and momentum, have been widely used to simulate nonlinear water waves in coastal and ocean engineering. One of the challenges of solving this class of partial differential equations is the numerical treatment of the dispersive terms described by the mixed third order derivative in space (twice) and time.

One objective of this paper is to develop 6th order coupled compact schemes for modified Boussinesq equations with analytical exact solution for comparison. These schemes demonstrate high order accuracy and fast convergence rates in both space and time. The idea in this paper could be used for solving other systems of coupled nonlinear equations.

This paper is outlined as below. In simulation approaches, the modified Boussinesq equations and their exact solutions corresponding to specially devised forcing functions are given. Then the solution ideas and procedures are presented, followed by schemes of sixth order accuracy developed for these equations. In simulation results, the solutions of all coupled variables and the convergence rates in space and time are demonstrated. In addition, the effect of long time integration is investigated to show numerical stability over time. Finally, conclusion summarises the novelty and uniqueness of the schemes developed in this paper, and future work is planned.

In terms of the velocity

To solve this system of equations, by the method of lines [

In order to compare results and investigate the rate of convergence, exact solutions to (

The major steps in finding numerical solution are elaborated below. These steps involve

Based on the initial

Based on

Based on

Now we focus on the spatial discretization and the specific high order compact schemes mentioned earlier. We partition the domain of interest,

Denote either

Denote

After the time integration with the Adams-Bashforth method,

We specially arranged

We compare our approximate (numerical) solutions with exact solutions for

Comparison of the approximate and exact solutions for

The performance of

Convergence rates for

To see the performance of

The performance of

(a) Convergence rates for

Now we examine the convergence rates for all three compact schemes together over 1,000 time steps after the initial conditions are imposed. During this period of time,

Similar to Figures

Convergence rates for combined schemes in space with (a)

After we examined the spatial convergence rates, now we demonstrate the convergence rate in time integration. To this end, we fix the spatial step size to

Convergence rates for combined schemes in time:

To examine the effect of long time integration, we fix the spatial resolution to

Long time integration over 168,000 time steps for

Next we reduce the time step size to

Long time integration over 220,000 time steps for

In this paper, coupled compact methods are used to solve nonlinear differential equations with third order mixed derivatives in space and time. We have developed sixth order compact schemes for numerical solutions to modified Boussinesq-type of equations, a coupled system of nonlinear differential equations with strong convective, and dispersive terms. The novelty of this work is summarized as below. First, the state variables include the original unknowns,

With the aid of convergence analyses in space and time, by fixing either the time step or the spatial step, the expected accuracy of these schemes is demonstrated. Long time integrations were performed to show the stability of these schemes.

In the subsequent work, we investigate the effectiveness of the proposed schemes for solutions with oscillatory signs, such as the velocity of water waves. Upwind treatment will be adopted for strong hyperbolic systems.

This work was supported by National Science Foundation under grants DMS-1115546 and DMS-1115527 and the state of Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22. The authors also appreciate their colleague Dr. Bernd Schroeder for providing constructive suggestions.

^{4}finite-difference approximations to operators of Navier-Stokes type