Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations
As is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semilinear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. Both the results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First a multi-frequency modulated Fourier expansion of the AAVF method is constructed and then two almost-invariants of the modulation system are derived.
Keywords: Semilinear wave equationsEnergy-preserving methodsMulti-frequency modulated Fourier expansionMomentum and actions conservation
where is a nonlinear and smooth real function with and is a real number satisfying . Similarly to the refs. [8, 9, 21], the initial values and for this equation are assumed to be bounded by a small parameter , which provides small initial data in appropriate Sobolev norms. Meanwhile, periodic boundary conditions are considered in this paper.
It is noted that several important quantities are conserved by the solution of (1). The total energy
is exactly preserved along the solution, where and the potential is of the form . The solution of (1) also conserves the momentum
where and are the Fourier coefficients in the series and , respectively. The harmonic actions
are conserved for the linear wave equation, where for . For the nonlinear case, it has been proved in [1, 9] that the actions remain constant up to small deviations over a long time period for smooth and small initial data and almost all values of .
It has now become a common practice that the consideration of qualitative properties in ordinary and more recently in partial differential equations is important when designing numerical schemes in the sense of structure preservation. In recent decades many numerical methods have been developed and researched for solving wave equations (see, e.g. [3, 4, 5, 10, 16, 17, 18, 24, 30, 33]). As one important aspect of the analysis, long-time conservation properties of wave equations or of some numerical methods applied to wave equations have been well studied and we refer the reader to [9, 8, 12, 13, 21]. All these analyses are achieved by the technique of modulated Fourier expansions, which was developed by Hairer and Lubich in  and has been frequently used in the long-term analysis (see, e.g. [7, 19, 22, 27, 29]). On the other hand, as an important kind of methods, energy-preserving (EP) methods have also been the subject of many investigations for wave equations. EP methods can exactly preserve the energy of the considered system. With regard to some examples of this topic, we refer the reader to [2, 6, 23, 25, 26, 28, 32]. Unfortunately however, it seems that the long-time behaviour of EP methods in other structure-preserving aspects has not been studied for wave equations in the literature, such as the numerical conservation of momentum and actions.
The main contribution of this paper is to rigorously analyse the long-time momentum and actions conservations of EP methods for wave equations. To our knowledge, this is the first research that studies the long-time behaviour of EP methods on wave equations by using modulated Fourier expansions. We organise the rest of this paper as follows. A full discretisation of the semilinear wave equation (1) by using spectral semi-discretisation in space and EP methods in time is given in Sect. 2. The main result of this paper is presented in Sect. 3 and a numerical experiment is carried out to support the theoretical result. The proof of the main result is given in detail in Sect. 4, where the modulated Fourier expansion of EP methods is constructed and two almost-invariants of the modulated system are studied. Some conclusions and further discussions are included in Sect. 5.
2 Full discretisation
2.1 Spectral semi-discretisation in space
We first discretise the wave equation in space by using a spectral semi-discretisation introduced in [8, 21]. Choose equidistant collocation points (for ) for the pseudo-spectral semi-discretisation in space and consider the real-valued trigonometric polynomials as an approximation for the solution of (1)
where and the prime indicates that the first and last terms in the summation are taken with the factor . We collect all the in a -periodic coefficient vector , which is a solution of the -dimensional system of oscillatory ODEs
where is diagonal with entries for and and denotes the discrete Fourier transform for It is seen that the system (3) is a finite-dimensional complex Hamiltonian system with the energy
where The actions (for ) and the momentum of (3) respectively read
where the double prime indicates that the first and last terms in the summation are taken with the factor . We are interested in real approximation (2) throughout this study and thus it holds that , and .
2.2 EP methods in time
It follows from (6) that
Clearly, the energy-preserving AAVF method does not exclude symmetry structure, and as is known that preserving the energy and symmetries of the systems at the discrete level is important for geometry integrators.
3 Main result and numerical experiment
In this paper, we take the following notations, which have been used in . For sequences of integers , , and a real , denote
Denote by the unit coordinate vector with the only entry at the -th position. For , the Sobolev space of -periodic sequences endowed with the weighted norm is denoted by . Moreover, we set
3.2 Main result
Before presenting the main result of this paper, the following assumptions are needed (see ).
It is assumed that the initial values of (3) are bounded by
with a small parameter .
The non-resonance condition is considered for a given stepsize :
If this is violated, we define a set of near-resonant indices
where is the truncation number of the expansion (14) which will be presented in the next section. We make the following assumption for this set. Suppose that there exist and a constant such that
We require the following numerical non-resonance condition
For a positive constant , consider another non-resonance condition
which leads to improved conservation estimates.
We are now in a position to present the main result of this paper.
Define the following modified momentum and actions, respectively
hold with a constant which depends on and , but not on and the time . If (12) is not satisfied, then the bound is weakened to .
The proof of this theorem will be presented in detail in Section 4 by using the technique of multi-frequency modulated Fourier expansions. It can be concluded from this theorem that the AAVF method has a near-conservation of a modified momentum and modified actions over long times. Although the result cannot be obtained for the momentum and actions , we note that and are no longer exactly conserved quantities in the semi-discretisation, which is seen from Theorem 1. Moreover, it will be shown in the next subsection that in comparison with the near-conservation of and , the modified momentum and modified actions are preserved better by AAVF method. This soundly supports the result of Theorem 2.
We have noticed that the authors in  analysed the long-time behaviour of a symmetric and symplectic trigonometric integrator for solving wave equations. It was shown in  that this integrator has a near-conservation of energy, momentum and actions in numerical discretisations. It is noted that the method studied in  cannot preserve the energy (4) exactly. However, from Proposition 1 and Theorem 2, it follows that the AAVF method not only preserves the energy (4) exactly but also has a near-conservation of modified momentum and actions over long times.
3.3 Numerical experiment
We now carry out a numerical experiment to show the numerical behaviour of AAVF method. The semilinear wave equation (1) with and is considered (see ) and its initial conditions are given by for . We consider the spatial discretisation with the dimension and consider applying midpoint rule to the integral appearing in the AAVF formula (5). It can be checked that the assumption (7) holds for . This problem is solved with the stepsize on and the relative errors of momentum/modified momentum and actions/modified actions against are shown in Figure 1. Here we use the following notations in the figures: and . It follows from the results that the modified momentum and modified actions are better conserved than the momentum and actions, which supports the results given in Theorem 2.
4 The proof of the main result
4.1 Preliminaries for the analysis
Define five operators by
where is the differential operator (see ). For these operators, the following results are clear.
The operator can be expressed in Taylor expansions as follows:
for and , where and . The Taylor expansions of are given by
for and . Moreover, for the operator with , we have
The following lemma is given in  which will be needed in the analysis of this paper.
(See .) For , one has For and , it is true that
where the sum is taken over satisfying . For , it is further true that
4.2 The outline of the proof
The proof relies on a careful research of a modulated Fourier expansion of the AAVF method (5). Assume that the conditions of Theorem 2 are true. For the numerical solution given by (5), we will construct the following truncated multi-frequency modulated Fourier expansion (with from (9))
where and , . For this modulated Fourier expansion, the following key points will be considered one by one in the rest of this section.
In Sect. 4.3 formal modulation equations for the modulation functions are derived.
In Sect. 4.4 we consider an iterative construction of the functions using reverse Picard iteration.
We then work with a more convenient rescaling and study the estimation of non-linear terms in Sect. 4.5.
Abstract reformulation of the iteration is presented in Sect. 4.6.
In Sect. 4.7 we control the size of the numerical solution by studying the bounds of modulation functions.
In Sect. 4.8 the bound of the defect is estimated.
We study the difference of the numerical solution and its modulated Fourier expansion in Sect. 4.9.
In Sect. 4.10 we show two invariants of the modulation system and establish their relationship with the modified momentum and modified actions.
Finally, the previous results that are valid only on a short time interval are extended to a long time interval in Sect. 4.11.
It is noted that the above procedure is a standard approach to studying the long-time behavior of numerical methods of Hamiltonian partial differential equations by using modulated Fourier expansions (see, e.g. [9, 8, 12, 13, 21]). The proof presented here closely follows these previous publications but with some modifications adapted to the AAVF method. The main differences in the analysis arise due to the implicitness of the AAVF method and the integral appearing in the method.
4.3 Modulation equations
Throughout the proof, denote by a generic constant which is independent of and .
In the light of the symmetry of the AAVF method and the following property
We look for a modulated Fourier expansion of the form
for the term . Then it is obtained that
In the same way, for , we have the following modulated Fourier expansion
which can be rewritten as
In what follows, we rewrite this equation by using the same way introduced in . We start with making the following notation. For a -periodic function , denote by the trigonometric interpolation polynomial to in the points . If is of the form then one has that by considering For a -periodic coefficient sequence , is referred to the trigonometric polynomial with coefficients , i.e., By using these new denotations, (18) becomes
Taylor expansion of the non-linearity at is given by 222It is noted that is used here.
where and the prime on the sum indicates that a factor is included in the appearance of with . Inserting this into (19), considering the th Fourier coefficient and comparing the coefficients of , we obtain
It is noted that the integral appearing here can be calculated exactly.
According to the Taylor expansion (13) of , the dominating term is for . If , then the dominating term is by considering the condition (8). If (8) is not true, the condition (10) ensures that the defect in simply setting is of size in an appropriate Sobolev-type norm. The above analysis and (20) determine the formal modulation equations of modulated functions .
For the modulation equations of , it follows from (5) that
By the definition of , this relation can be expressed as
In terms of the Taylor series of , the relationship between and can be established:
for which gives the modulation equations of .
4.4 Reverse Picard iteration
Denote by the th iterate. For , we consider the iteration procedure as follows:
For and satisfying the non-resonant (8), the iteration procedure is of the form
In these iterations it is assumed that and for . There is an initial value problem of first-order ODEs for (for ) and algebraic equations for with at each iteration step. The starting iterates () are chosen as for , and