Simulation of blow-up solutions to the generalized KdV equations by moving collocation methods
- Zhiqiang Zhou^{1}Email author and
- Xiaodan Wu^{1}
Received: 1 November 2015
Accepted: 26 January 2016
Published: 18 February 2016
Abstract
The aim of this paper is to simulate the blow-up solutions to generalized Korteweg-de Vries (KdV) equations. The KdV equations are discretized with the use of a quintic Hermite collocation method based on the moving meshes generated by solving moving mesh partial differential equations (MMPDEs). Theoretical analyses are, respectively, conducted to determine the critical parameters in MMPDEs such that the generated meshes can catch up with the blow-up profiles and to show the effectiveness of the generated moving meshes. Lastly, a variety of examples are implemented to confirm our analysis and show the efficiency of the method.
Keywords
MSC
1 Introduction
To compute such a type of singularities sufficiently and effectively, and hence to mimic the asymptotic behavior of the solution as \(t\rightarrow T\), the numerical method employed is required to adapt the spatial meshes to the evolving singularities. In order to achieve this, Bona et al. employ the h-adaptation technique [5, 6]: a spatial translation is used to keep the blow-up peak appearing near the fixed point \(x=0.5\) and local mesh refinement is conducted recursively right around this point. The demerit of this local refinement technique is that the computational cost becomes larger and larger as the blow-up solution evolves. Moreover, the use of interpolations in the time integration does not keep the truncation errors under control. In this paper, based on the moving mesh method, we provide a more reliable simulation method along with an in-depth analysis.
It can be observed that, for both MMPDE5 and MMPDE6, the mesh trajectory speeds depend on M and τ. Moreover, for any choice of M and τ, the two MMPDEs can generate smooth meshes as long as the time scale is taken to be sufficiently small. Therefore, the key issues we need to deal with in the simulation of blow-up are: 1. how to choose M and τ so that the mesh trajectory speed is as fast as that in (4); 2. how small the time scale should be in order that the MMPDEs can generate smooth meshes and resolve the dramatically increase in the blow-up solution.
Among the earlier literature which uses MMPDE5 and MMPDE6 to generate moving meshes in the simulation of blow-up solutions, Huang et al. [13] chose the monitor parameters M and τ according to what is called ‘the dominance of equidistribution’, and introduced dimensional analysis. Different from [13], our paper uses the criterion that the mesh trajectory speed satisfies (4) for determining the parameters M and τ, as stated above, and presents a theoretical analysis to show the efficiency of the generated meshes.
We organize the paper as follows: in Section 2, the conservative moving collocation method of the fifth order is proposed to discretize (1); in Section 3, through dimensional analysis, we discuss the choices of the monitor function M and the parameter τ in the MMPDEs; in Section 4, the efficiency of the moving mesh collocation is analyzed; in Section 5, numerical examples are carried out to confirm our analysis and to simulate the blow-up.
2 The conservative moving collocation method
3 Choice of monitor functions
Moreover, notice that blow-up only occurs in the solution to GKdV equations with \(p>4\). Even if we can obtain a satisfactory mesh trajectory speed by choosing a monitor function as \(|u|^{\gamma_{1}}\) with \(\gamma_{1}\geq\frac{3p}{2}\), such a large power of the solution will generally result in over-concentration of the mesh points within the blow-up region and cause the simulation to break down. The same problem also occurs when the monitor function is chosen to be (15) or (16). In view of this, we shall only consider the case that \(\gamma_{1}\) and \(\gamma_{2}\) have small values. In particular, we fix \(\gamma_{1}=\gamma_{2}=1\) while we vary the value of τ to obtain a satisfactory mesh speed in the rest of the paper.
Monitor function \(M=\vert {\frac {\partial{u}}{\partial{x}}} \vert ^{\gamma_{2}}\):
4 An analysis
In this section, we carry out a careful analysis to verify the validity of MMPDE5 in simulating blow-up solutions when the monitor function M, τ are chosen as in {(16), (26)}, respectively. Similar analysis can be performed for MMPDE5 with the choice of M, τ as in {(14), (23)} or as in {(15), (25)}. The analysis can also be conducted for MMPDE6 with the choices of M, τ above mentioned.
5 Numerical examples
In this section, we demonstrate the efficiency and accuracy of the proposed moving collocation method for solving GKdV equations. Example 5.1 is chosen to illustrate the rate of convergence while Examples 5.2 and 5.3 show the capability of our method to accurately capture the two important features of the GKdV equations - solitary waves and blow-up.
We mention here that the numerical solution of MMPDE6 is approximated by using central difference method for the spatial derivatives and backward Euler method for the time derivative.
5.1 Convergence rates
We calculate the following example to show the convergence rates of the conservative collocation methods based on fixed and moving meshes.
Example 5.1
Results for the uniform mesh in Example 5.1
N | 40 | 70 | 100 | 130 | 160 |
---|---|---|---|---|---|
Error | 1.18e−3 | 1.44e−4 | 3.56e−5 | 1.24e−5 | 5.19e−6 |
Rate | - | 3.75 | 3.92 | 4.03 | 4.19 |
Results for the moving mesh in Example 5.1
N | 40 | 70 | 100 | 130 | 160 |
---|---|---|---|---|---|
Error | 1.02e−4 | 1.34e−5 | 3.30e−6 | 1.04e−6 | 4.06e−7 |
Rate | - | 3.63 | 3.94 | 4.37 | 4.57 |
From Tables 1 and 2, we may see that the convergence rates for the conservative collocation methods are 4. The convergence order of the moving collocation method with fifth-order Hermite polynomial basis is expected to be 6; however, this is only true for some special types of PDE. For the KdV equations which are nonlinear, it is indeed not guaranteed that the moving collocation scheme can attain the sixth-order convergence. The numerical tests show that the conservative collocation methods are of high-order schemes for both fixed (uniform) meshes and moving meshes. Moreover, the error for the moving mesh is smaller than that for fixed (uniform) mesh. In fact, in the literature only one paper by Ma et al. [12] analyzes the convergence order of moving collocation method for linear second-order PDEs - a very simple PDE; however, it is not possible to prove the convergence rates for nonlinear KdV equations.
5.2 Capture of solitary waves and blow-up
In this section we use two examples to show the capability of our method to accurately capture the two important features of the GKdV equations - solitary waves and blow-up.
Example 5.2
Example 5.3
From the above tests, we can observe that the solitary waves move from one side to the other with the peak value increasingly and eventually ends up with a blow-up. Also we observe that there are oscillations in the non-blow-up regions. That means that there are also waves of small amplitude in the non-blow-up regions.
6 Conclusions
Applied to second- and fourth-order PDEs [9, 10], moving collocation methods show a high order of convergence and capabilities of capturing blow-up phenomena. In this paper, we develop the moving collocation method for third-order PDEs - the KdV equations. The method can be easily generalized to third-order PDEs of other types. The fully discrete moving collocation method proposed is shown to be fourth-order convergent in space and it is then employed to simulate the blow-up solutions to the generalized KdV equations. In the simulation of the blow-up solutions, we use MMPDEs to generate the moving meshes: we first assess how fast the mesh trajectory speed is required to be according to the structure of the blow-up solutions, and then determine the parameters in the MMPDEs through dimensional analysis. The theoretical analysis and the numerical experiments show that the method can accurately capture the two important features of the GKdV equation - solitary waves and blow-up.
Declarations
Acknowledgements
The authors are grateful to Prof. J Ma for the helpful suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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