Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

07/18/2019 ∙ by Qian Zhang, et al. ∙ USTC 0

In the present paper, we consider the discontinuous Galerkin (DG) methods for solving short pulse (SP) type equations. The short pulse equation has been shown to be completely integrable, which admits the loop-soliton, cuspon-soliton solutions as well as smooth-soliton solutions. Through hodograph transformations, these nonclassical solutions can be profiled as the smooth solutions of the coupled dispersionless (CD) system or the sine-Gordon equation. Thus, DG methods can be developed for the CD system or the sine-Gordon equation to simulate the loop-soliton or cuspon-soliton solutions of the SP equation. The conservativeness or dissipation of the Hamiltonian or momentum for the semi-discrete DG schemes can be proved. Also we modify the above DG schemes and obtain an integration DG scheme. Theoretically the a-priori error estimates have been provided for the momentum conserved DG scheme and the integration DG scheme. We also propose the DG scheme and the integration DG scheme for the sine-Gordon equation, in case the SP equation can not be transformed to the CD system. All these DG schemes can be adopted to the generalized or modified SP type equations. Numerical experiments are provided to illustrate the optimal order of accuracy and capability of these DG schemes.

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1 Introduction

In this paper, we mainly study the classic short pulse (SP) equation derived by Schäfer and Wayne in [25]

(1.1)

The SP equation models the propagation of ultra-short light pulses in silica optical fibers. Here, is a real-valued function which represents the magnitude of the electric field. It is well-known that the cubic nonlinear Schrödinger (NLS) equation derived from the Maxwell’s equation can describe the propagation of pulse in optical fibers. Two preconditions of this derivation need to be satisfied: First, the response of the material attains a quasi-steady-state and second that the pulse width is as large as the oscillation of the carrier frequency. And now we can create very short pulses by the advanced technology and the pulse spectrum is not narrowly localized around the carrier frequency, that is, when the pulse is as short as a few cycles of the central frequency. Therefore, we use SP equation to approximate the ultra-short light pulse. And numerical experiments made in [8] show as the pulse shortens, the accuracy of the SP equation approximated to Maxwell’s equation increases, however, the NLS equation becomes inaccuracy for the ultra-short pulse. If the pulse is as short as only one cycle of its carrier frequency, then the modified short pulse equation in [28] is used to describe the propagation of pulse in optical fibers. Similar to the extension of coupled nonlinear Schrödinger equations from NLS equations, it is necessary to consider its two-component or multi-component generalizations for describing the effect of polarization or anisotropy [10, 22, 29]. For birefringent fibers, the authors in [11, 13] also introduced some extensions of the SP equation to describe the propagation of ultra-short pulse. We will introduce these extensions specifically in Section 3.

Integrable discretizations of short pulse type equations have received considerable attention recently, especially the loop-soliton, antiloop-soliton and cuspon-soliton solutions in [11, 12, 13, 14, 29]. The authors linked the short pulse type equations with the coupled dispersionless (CD) type systems or the sine-Gordon type equations through the hodograph transformations. The key of the discretization is an introduction of a nonuniform mesh, which plays a role of the hodograph transformations as in continuous case. In this paper, we aim at solving the loop-soliton, cupson-soliton solutions of the short pulse type equations as well as smooth-soliton solutions. Through the hodograph transformation which was proposed in [26],

(1.2)

we can establish the link between the SP equation (1.1) and the CD system [17],

(1.3)

There exists some short pulse type equations which are failed to be transformed into CD systems. Therefore, we consider an alternative approach by introducing a new variable and define another hodograph transformation,

(1.4)

which connects the short pulse equation (1.1) with the sine-Gordon equation [28]

(1.5)

For the CD system or the sine-Gordon equation, we develop the discontinuous Galerkin (DG) schemes to obtain the high-order accuracy numerical solution or . Consequently, a point-to-point profile of loop-soliton, cuspon-soliton solutions of the SP equation can be obtained, which are shown by the numerical experiments in Section 4.

The DG method was first introduced in 1973 by Reed and Hill in [24] for solving steady state linear hyperbolic equations. The important ingredient of this method is the design of suitable inter-element boundary treatments (so called numerical fluxes) to obtain highly accurate and stable schemes in many situations. Within the DG framework, the method was extented to deal with derivatives of order higher than one, i.e. local discontinuous Galerkin (LDG) method. The first LDG method was introduced by Cockburn and Shu in [7] for solving convection-diffusion equation. Their work was motivated by the successful numerical experiments of Bassi and Rebay [2] for compressible Navier-Stokes equations. Later, Yan and Shu developed a LDG method for a general KdV type equation containing third order derivatives in [39], and they generalized the LDG method to PDEs with fourth and fifth spatial derivatives in [40]. Levy, Shu and Yan [19] developed LDG methods for nonlinear dispersive equations that have compactly supported traveling wave solutions, the so-called compactons. More recently, Xu and Shu further generalized the LDG method to solve a series of nonlinear wave equations [34, 35, 36, 37]. We refer to the review paper [33]

of LDG methods for high-order time-dependent partial differential equations.

Most recently, a series of schemes which called structure-preserving schemes have attracted considerable attention. For some integrable equations like KdV type equations [9, 18, 20, 42], Zakharov system [31], Schrödinger-KdV system [32], Camassa-Holm equation [41], etc., the authors proposed various conservative numerical schemes to “preserve structure”. These conservative numerical schemes have some advantages over the dissipative ones, for example, the Hamiltonian conservativeness can help reduce the phase error along the long time evolution and have a more accurate approximation to exact solutions for KdV type equations [42]. The CD system and the generalized CD system are integrable, thus they have an infinite number of conserved quantities [16]. For CD system, the following two invariants

(1.6)

are corresponding to the Hamiltonian of the SP equation [3, 4] via the hodograph transformation,

(1.7)

In this paper, we first construct conserved DG scheme for the SP equation directly. And for the loop-soliton and cuspon-soliton solutions, the , conserved DG schemes for CD system are developed respectively, to profile the singular solutions of the SP equation. Also we modify the above DG schemes and propose an integration DG scheme which can numerically achieve the optimal convergence rates for , and . Theoretically, we prove that the conserved DG scheme has the optimal order of accuracy for and in norm. The integration DG scheme can be proved the optimal order of accuracy for in norm and the suboptimal order of accuracy for in norm. All these DG schemes can be adopted to the generalized or modified SP type equations.

The rest of this paper is organized as follows: In Section 2, we develop the DG schemes for the SP equation directly, and via the hodograph transformations for the CD system and the sine-Gordon equation. Some notations for simplifying expressions are given in Section 2.1. In Section 2.2, we first propose the conserved DG scheme for the SP equation. To simulate the loop-soliton or cuspon-soliton solutions of the SP equation, the , conserved DG schemes and the integration DG scheme are constructed for the CD system which links the SP equation by the hodograph transformation. Meanwhile, the a priori error estimates for conserved DG and integration DG schemes are also provided. Moreover, we develop two kinds of DG schemes for the sine-Gordon equation to introduce another resolution for the SP equation in Section 2.3. Section 3 is devoted to summarize the generalized short pulse equations and introduce the corresponding conserved quantities briefly. Several numerical experiments are listed in Section 4, including the propagation and interaction of loop-soliton, cuspon-solution, breather solution of the short pulse type equations. We also show the accuracy and the change of conserved quantities in Section 4. Finally, some concluding remarks are given in Section 5.

2 The discontinuous Galerkin discretization

In this section, we present the discontinuous Galerkin discretization for solving the short pulse type equations. In order to describe the methods, we first introduce some notations.

2.1 Notations

We denote the mesh on the spatial by for , with the cell center denoted by . The cell size is and . The finite element space as the solution and test function space consists of piecewise polynomials

where denotes the set of polynomial of degree up to defined on the cell . Note that functions in are allowed to be discontinuous across cell interfaces. We also denote by the and the values of at , from the left cell and the right cell respectively. And the jump of is defined as , the average of as . To simplify expressions, we adopt the round bracket and angle bracket for the inner product on cell and its boundary

for one dimensional case.

For the spatial variable , we denote the mesh by for . Similar to the notations on the mesh , we have , and . Without misunderstanding, we still use and denote the values of at , from the left cell and the right cell respectively.

2.2 The short pulse equation

Recall the short pulse equation

(2.1)

where is a real-valued function, denotes the time coordinate and is the spatial scale. Through the hodograph transformation, it can be converted into a coupled dispersionless (CD) system

(2.2a)
(2.2b)

where denotes the time coordinate, and is the spatial scale, . The hodograph transformation is defined by

(2.3)

And the parametric representation of the solution of the short pulse equation (2.1) is

(2.4)

where is a real constant. Since the short pulse equation and the equivalent CD system are completely integrable and hence they have an infinite number of conservation laws. The first two invariants of the SP equation are described by

(2.5)

and the corresponding conservation laws for the CD system are

(2.6)

2.2.1 conserved DG scheme

To construct the discontinuous Galerkin method for the SP equation directly, we rewrite the SP equation (2.1) as a first order system:

(2.7a)
(2.7b)
(2.7c)

Then the local DG scheme for equations (2.7c) is formulated as follows: Find such that, for all test functions and

(2.8a)
(2.8b)
(2.8c)

where . The “hat” terms in the scheme are the so-called “numerical fluxes”, which are functions defined on the cell boundary from integration by parts and should be designed based on different guiding principles for different PDEs to ensure the stability and local solvability of the intermediate variables. To ensure the scheme is conserved, the numerical fluxes we take are

(2.9a)
(2.9b)

where .

Proposition 2.1.

(Energy conservation) The DG scheme (2.8c) with the numerical fluxes (2.9b) for the short pulse equation (2.1) satisfies the energy conservativeness

(2.10)
Proof.

For the equation (2.8b), we take the time derivative and get

(2.11)

Since (2.11), and (2.8a)-(2.8c) hold for any test functions in , we can choose

(2.12)

and it follows that

(2.13)
(2.14)
(2.15)

To eliminate extra terms, we take test functions in (2.8a), in (2.11), and then obtain

(2.16)
(2.17)

With these choices of test functions and summing up the five equations in (2.13)-(2.17), we get

(2.18)

Now the equation (2.18) can be rewritten into following form

(2.19)

where the numerical entropy flux is given by

(2.20)
(2.21)

and the extra term is defined as

(2.22)

which vanishes due to choices of the conservative numerical fluxes (2.9b). Summing up the cell entropy equalities (2.19) with periodic or homogeneous Dirichlet boundary conditions, implies that

(2.23)

Thus, the DG scheme (2.8c) for the short pulse equation is conserved. ∎

The conserved DG scheme resolves the smooth solutions for the short pulse equation well, as we show in Section 4. Numerically, conserved DG scheme can achieve - order of accuracy for even , and -

order of accuracy for odd

. However, for the loop-soliton and cuspon-soliton solutions, this scheme can not be used due to the singularity of solutions. So we introduce the DG schemes via hodograph transformations in the following sections.

2.2.2 conserved DG scheme

As we have mentioned, the short pulse equation can be converted into the coupled dispersionless (CD) system through the hodograph transformation. To construct the local discontinuous Galerkin numerical method for the CD system, we first rewrite (2.2b) as a first order system

(2.24a)
(2.24b)
(2.24c)
(2.24d)

Then we can formulate the LDG numerical method as follows: Find , , , such that

(2.25a)
(2.25b)
(2.25c)
(2.25d)

for all test functions , , , and . To guarantee the conservativeness of , we adopt the central numerical fluxes

(2.26)

Numerically we will see that the optimal - order of accuracy can be obtained for , when is even, however, the numerical solutions , have - order of accuracy when is odd. If we modify numerical fluxes as below:

(2.27)

then the scheme is dissipative on with the appropriate parameters in Proposition 2.2 and the optimal order of accuracy can be achieved numerically for this dissipative scheme.

Proposition 2.2.

( conservationdissipation) The semi-discrete DG numerical scheme (2.25d), (2.26) can preserve quantity spatially. And the scheme (2.25d) with (2.27) composes a dissipative DG scheme on if the parameters in (2.27) satisfy the conditions

(2.28)
Proof.

First, we take time derivative of equation (2.25c), and the test functions are chosen as . Then we have

(2.29)
(2.30)
(2.31)
(2.32)

Summing up all equalities (2.29)-(2.32), we obtain

(2.33)

which can be written as

(2.34)

where the numerical entropy fluxes are given by

(2.35)

and the extra term is

(2.36)

Therefore the choices of in (2.27) concern the conservativeness of the DG scheme. According to the parameters , we give below two cases:

conserved DG scheme : (2.37)
dissipative DG scheme : (2.38)

Summing up the cell entropy equalities (2.34) and (2.37), (2.34) and (2.38), respectively, then we get

(2.39)

In the numerical test Example 4.2, it shows that the dissipative scheme with parameters (2.38) can achieve the optimal convergence rate for both and no matter is odd or even. However, the order of accuracy for the conserved DG scheme is - for odd , - for even . The choices of these parameters are not unique, but the above numerical fluxes in the dissipative scheme can minimize the stencil as in [42].

2.2.3 conserved DG scheme

In this section, we construct another discontinuous Galerkin scheme which preserves the quantity of the CD system (2.2b) which links the Hamiltonian of the short pulse equation through the hodograph transformation. First, we rewrite the CD system as a first order system

(2.40a)
(2.40b)
(2.40c)

Then the semi-discrete LDG numerical scheme can be constructed as: Find , , such that

(2.41a)
(2.41b)
(2.41c)

for all test functions , , and . The numerical flux is taken as . Numerically, the optimal - order of accuracy can be obtained for both , . If we take , then the accuracy is - order for , when is even, and - order of accuracy when is odd.

Proposition 2.3.

( conservation) The semi-discrete DG numerical scheme (2.41c) can preserve the quantity spatially.

Proof.

By taking the test functions in (2.41c), we obtain

Summing up over all , it implies that

In what follows, we prepare to give the a priori error estimate for the conserved DG scheme. The standard projection of a function with continuous derivatives into space , is denoted by , that is, for each

(2.42)

and the special projections into satisfy, for each

(2.43)
(2.44)

For the projections mentioned above, it is easy to show [6] that

(2.45)

where or , and the positive constant only depends on . There is an inverse inequality we will use in the subsequent proof. For , there exists a positive constant (we call it the inverse constant), such that

(2.46)

where .

First, we write the error equations of the conserved DG scheme as follows:

(2.47)
(2.48)
(2.49)

and denote

(2.50)

To deal with the term , we need to establish a relationship between and in following lemma.

Lemma 2.4.

The are defined in (2.50), then there exists a positive constant independent of but depending on inverse constant and Poincaré constant , such that

(2.51)
Proof.

By Poincaré -Friedrichs inequality in Chapter of [5], we have

where

The inequality (4.17) in [30] gives