In this paper, we consider the initial value problem of the Ostrovsky-Vakhnenko equation
which can be viewed as a particular limit of the generalized Korteweg-de Vries (KdV) equation
This equation (1.2) was derived in  as a model to describe the small-amplitude long waves on a shallow rotating fluid. Concerning the structure of this equation, it has a purely dispersive term. Although it has the same nonlinear term of the KdV equation, the dispersive terms are different. When , there is no high-frequency dispersion. In , Vakhnenko uses (1.1) to describe high frequency waves in a relaxing medium. In a series of papers [25, 18, 26], its integrability was established by deriving explicit solutions. It is known under different names in some literatures, such as the reduced Ostrovsky equation, the Ostrovsky-Hunter equation, the short-wave equation and the Vakhnenko equation, in this paper, we call (1.1) as Ostrovsky-Vakhnenko (OV) equation. The Ostrovsky-Vakhnenko equation has two properties that appear to be generic,
Travelling waves that exist only up to a maximum limiting amplitude,
Limiting waves that have corners, i.e., a slope discontinuity.
The OV equation has peakon, shock and wave breaking phenomena even for smooth initial conditions in finite time. In [12, 17, 13], the authors have discussed the condition for wave breaking. Some exact solutions including periodic solution, and solitary traveling wave solution are investigated in [13, 19, 20]. Well-posedness results can be found in [11, 16, 27]. Through the hodograph transformation, [8, 9] provide the cuspon and loop soliton solutions for the generalized OV system. Several numerical methods are proposed for the OV equation such as Fourier pseudo-spectral methods  and a finite difference scheme based on the Engquist-Osher scheme [7, 22]. Additionally, rigorous numerical analysis of the OV equation is concluded by Coclite, Ridder and Risebro in [7, 22], including the convergence results and the existence of entropy solution. In , bi-Hamiltonian structure of the OV equation is confirmed, i.e., the OV equation has infinite conservative quantities, in which we investigate energy and Hamiltonian as
The development of our numerical schemes are based on these two conservative quantities. As the conservative methods for KdV equation [3, 14, 40], Zakharov system , Schrödinger-KdV system , short pulse equation , etc., various conservative numerical schemes are proposed to “preserve structure”. Usually, the conservative schemes can help reduce the phase error along the long time evolution.
The DG method was first introduced in 1973 by Reed and Hill in  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 several situations. Within the DG framework, the method was extended 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  for solving convection-diffusion equation. Their work was motivated by the successful numerical experiments of Bassi and Rebay  for compressible Navier-Stokes equations. Later, Yan and Shu developed an LDG method for a general KdV type equation containing third order derivatives in , and they generalized the LDG method to PDEs with fourth and fifth spatial derivatives in . Levy, Shu and Yan  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 [28, 29, 30, 31, 39]. We refer to the review paper 
of LDG methods for high-order time-dependent partial differential equations.
Here, we adopt the DG method as a spatial discretization to construct high order accurate numerical schemes for the OV equation. For general solutions, the Hamiltonian conservative DG scheme and the energy stable schemes that contain the DG scheme and the integration DG scheme are developed. The energy stable schemes work for the smooth, peakon and shock solutions. The Hamiltonian conservative DG scheme can handle the smooth, peakon solutions and preserve the Hamiltonian spatially. The stability and conservation refer to the semi-discrete properties. For the time discretization, we use the so called total variation diminishing (TVD) or strong stability preserving (SSP) Runge-Kutta methods in [23, 10]. For some singular soliton solutions, we utilize the hodograph transformation to transform the OV equation to a coupled dispersionless (CD) type system, and then develop the DG scheme for the transformed CD system.
The paper is organized as follows. In Section 2, we directly construct two energy stable and Hamiltonian conservative DG schemes for the OV equation. We provide proofs of stability and Hamiltonian conservation. Suboptimal error estimates of the two energy stable schemes are also proved in this section. For some singular soliton solutions, including loop and cuspon solitons, we transform the OV equation to the CD system via the hodograph transformation in Section 3. Subsequently, two DG schemes are developed for the CD system to obtain the numerical solutions for the OV equation indirectly. Some numerical experiments are presented in Section 4 to show the results of approximation. This paper is concluded in Section 5.
2 The DG methods for the Ostrovsky-Vakhnenko equation
In this section, we develop two kinds of DG methods for the OV equation (1.1), including energy stable schemes and the Hamiltonian conservative DG scheme.
We denote the mesh 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 . Notably, the functions in are allowed to be discontinuous across cell interfaces. The values of at are denoted by and , from the left cell and the right cell , respectively. Additionally, the jump of is defined as , the average of as .
After the hodograph transformation, the spatial variable change into from . We denote the mesh by for . As the same definition on variable , we have .
To simplify expressions, we adopt the round bracket and angle bracket for the inner product on cell and its boundary
for one dimensional case.
2.2 The Energy stable schemes
In this section, we develop two DG schemes with energy stability, and for smooth solutions, suboptimal order of accuracy - is proved for these two DG schemes. To distinguish other DG schemes in this paper, we call it the energy stable DG scheme and energy stable integration DG scheme for the OV equation (1.1).
2.2.1 The DG scheme for the OV equation
First, we divide the OV equation into a first order system
For the Dirichlet boundary problem, the fixed boundary condition for is adopted,
For the periodic boundary problem, the zero mean condition is adopted.
: The energy stable DG scheme is formulated as follows: Find the numerical solutions , for all test functions , such that
where . The “hat” terms in (2.4b) 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 introduce some dissipation of energy, we adopt the dissipative numerical fluxes as
The flux we consider here is the Lax-Friedrichs flux, which is regarded as a dissipative flux. The numerical flux depends on the sign of the parameter . When is positive, is taken as , otherwise, . Numerically, the DG scheme with numerical fluxes (2.5), (2.6) can achieve - order of accuracy.
: Alternatively, we can integrate the equation directly instead of the DG scheme (2.4b). Therefore, the energy stable integration DG scheme is defined as: Find the numerical solutions , , for all test functions , such that
The equation (2.7b) can also be replaced by
which depends on the boundary condition of . For the constraint of numerical solution , we will give a more specific explanation in next section.
2.2.2 Algorithm flowchart
In this part, we give some details related to the implementation of our numerical Scheme 1 and Scheme 2. We can see that the equation (2.4a), (2.7a) are exactly the same. The main difference between Scheme 1 and Scheme 2 lies in (2.4b) and (2.7b), respectively, which we will explain in Step 1.
In Scheme 1: From the equation (2.4b), we have the following matrix form,
Here,and , respectively. The size of matrix is , is the number of spatial cells and is the degree of the approximate space . However, if is periodic, the matrix is under-determined and the rank of is . Therefore, as a replacement, the zero mean condition helps determine a unique solution.
In Scheme 2: Under the fixed boundary condition of , we choose in (2.7b) or in (2.8). And then can be solved cell by cell. For the zero mean condition, we also choose . According to the equation (2.7b), we can get on each cell . Subsequently, we check our zero mean condition by calculating the value of . Generally, will not be zero. Thereafter, a modification is done for the value of ,
Subsequently, we obtain a numerical solution that satisfies the condition .
Step 2 : Substituting into the equation (2.4a), we have
By choosing a suitable ODE solver, such as Runge-Kutta time discretization method, we will finally implement these two numerical schemes.
2.2.3 stability of the energy stable schemes
The stability of Scheme 1 and Scheme 2 are presented in Proposition 2.1 and 2.2, respectively. This is the reason why we call Scheme 1 as the energy stable DG scheme, and Scheme 2 as the energy stable integration DG scheme.
( stability for Scheme 1)
We take the test function , in scheme (2.4b), thereafter, we obtain
After applying summation of the above-mentioned two equations, we have
where the numerical entropy flux is
and the extra term is given by
Thereafter, we find that the whole term is non-negative. Summing up the cell entropy equalities (2.14) with the periodic boundary condition or homogeneous Dirichlet boundary condition, we have the energy stability as
i.e., energy stability of the DG scheme (2.4b) for the OV equation. ∎
( stability for Scheme 2)
The semi-discrete DG scheme (2.7b) is an energy stable scheme, i.e.,
We take test function in (2.7a),
Additionally, following the idea of Proposition 2.1, for the nonlinear term , we can have a stable property. There is an extra term required to be estimated. The Scheme 2, which is also called the integration DG method, is based on , then
Due to the continuity of and the periodic or homogeneous Dirichlet boundary condition, we obtain the result of stability after summing up the equation (2.19) over all cells,
2.2.4 Error estimates of the energy stable schemes
In this section, the a-priori error estimate of Scheme 1 (2.4b) and Scheme 2 (2.7b) will be stated. Referring to the procedure in [38, 32], we will give the brief proofs in the subsequent descriptions. Without loss of generality, we let in this part.
First, we make some preparations for error estimate by giving necessary assumptions, projection and interpolation properties. The standardprojection of a function with continuous derivatives into space , is denoted by , i.e., for each
and the special projections into satisfy, for each
For the projections mentioned above, it is easy to show  that
where or , and the positive constant only depends on .
We will use an inverse inequality in the subsequent proofs. For , there exists a positive constant (we call it the inverse constant), such that
Additionally, to deal with the nonlinearity of the flux , we make a priori assumption that, there holds
for small enough . Under this assumption, the error of norm satisfies
where or .
For the Scheme 1, we have below theorem to demonstrate the result of convergence for smooth exact solutions.
It is assumed that the OV equation (2.2) with periodic boundary condition has a sufficiently smooth exact solution . The numerical solution satisfies the semi-discrete DG scheme (2.4b) with flux (2.5),(2.6). For regular partitions of , and the finite element space with , there holds the following error estimate for small enough
First, we give the error equation between the exact solution and numerical solution,
for all test functions . Thereafter, we define two bilinear forms
After applying summation over all cells , the error equation is expressed by
and taking test functions , we have
Using Young’s inequality and interpolation properties (2.28), we get
Utilized Gronwall’s inequality, the equation becomes
For the Scheme 2, we also state the following error estimate for smooth exact solutions.
Similarly, we give the error equations,
for any test function . Thereafter, we define another bilinear form
After applying summation over all cells , the error equations are expressed by
We define notations
where , it satisfies . By Poincaré inequality, the error of is controlled by ,
With the test function , we have
For bilinear form , the following equation holds by projection properties (2.22)
2.3 The Hamiltonian conservative DG scheme
In this section, we construct another DG scheme, which can preserve the Hamiltonian spatially. Therefore, we call it the Hamiltonian conservative DG scheme for the OV equation.
We rewrite the OV equation as another first order system
The Hamiltonian conservative DG scheme is defined as: Find numerical solutions , for all test functions , such that
The numerical fluxes are taken as
The difference between the Hamiltonian conservative DG scheme and energy stable DG scheme in section 2.2 is an projection (2.49b). Therefore, the implementation of algorithm is similar to Section 2.2.2, the flowchart is omitted here.
First, we take the time derivative of the equation (2.49c) to obtain
To eliminate the extra term , we take the test function in equation (2.52), and obtain