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Dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly


Introduction
The Fornberg-Whitham type equation we study in this paper is given by x ) −1 u x = 0, I ∈ [a, b], t > 0 (1. 1) or its equivalent form by operating (1 − ∂ 2 x ) on (1.1). We consider the nonlinear term f (u) = 1 p u p , where p ≥ 2 is an integer. When the parameter p = 2, the equation (1.1) becomes the Fornberg-Whitham equation derived in [21] as a nonlinear dispersive wave equation. There are three conservative quantities for the Fornberg-Whitham type equation where the quantity E 0 is called mass, and E 2 is energy.
Many mathematical properties of the Fornberg-Whitham equation have been discussed, this equation was first proposed for studying the qualitative behavior of wave breaking in [21]. Some investigation of wave breaking conditions can be found in [14,12].
Note that the Fornberg-Whitham equation is also called Burgers-Poisson equation in [8].
There has been lots of work focusing on finding the traveling wave solutions in [42,41].
It admits a wave of greatest height, as a peaked limiting form of the traveling wave solution [7]. Recently, some well-posedness results are proposed in [10,11]. There are not many numerical schemes for the Fornberg-Whitham type equation. In [13], the finite difference method is adopted to solve the shock solution. The authors did some valuable numerical analysis by the discontinuous Galerkin method in [16], in which the comparisons has been made between the conservative scheme and dissipative scheme, as well as theoretical analysis.
The discontinuous Galerkin (DG) method was first introduced by Reed and Hill in 1973 [19] for solving steady-state linear hyperbolic equations. The key point 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 local discontinuous Galerkin (LDG) method can be obtained by extending to handle derivatives of order higher than one. The first LDG method was introduced by Cockburn and Shu in [6] for solving the convection-diffusion equation. Their work was motivated by the successful numerical experiments of Bassi and Rebay [1] for compressible Navier-Stokes equations. The LDG methods can be applied in many equations, such as KdV type equations [32,26,27,28,15,39], Camassa-Holm equations [24,35], Degasperis-Procesi equation [31], Schrödinger equations [25,23], and more nonlinear equations or system [33,24,30,34,17].
There are also many conservative DG schemes that are proposed to "preserve structure", such as KdV equation [3,15,39], Zakharov system [22], Schrödinger-KdV system [23], short pulse equation [40], etc. Usually, the structure preserving schemes can help reduce the shape error of waves along with long time evolution. For example in [3,16,15,39], compared with dissipative schemes, the energy conservative or Hamiltonian conservative numerical schemes for the KdV equation have less shape error or amplitude damping for long time approximations, especially in the low-resolution cases.
In this paper, we adopt the LDG method as a spatial discretization to construct high order accurate numerical schemes for the Fornberg-Whitham type equations. Through the two equivalent forms of this type of equation, we develop dissipative and conservative schemes, respectively. The corresponding conservative quantities can be proved as dissipative or conservative for the semi-discrete schemes. For the dissipative schemes, the theoretical results are confirmed as the (k + 1)-th order of accuracy if p is odd, and (k + 1 2 )-th order of accuracy if p is even.
Here and in what follows, k is the polynomial degree of the finite element space. The proof for the dissipative schemes is motivated by the work on conservation laws [37,38]. Since the numerical fluxes are different from the work [37,38], there exist technical obstacles to derive the a priori error estimates due to the lack of control on some jump terms at interfaces. The error estimate results for the Fornberg-Whitham type equations can not achieve the same optimal order as in [37,38] for the upwind flux. For the conservative schemes, we prove the k-th order of accuracy in a different way from [3]. Numerically, the dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly. The two proposed LDG schemes vary slightly on efficiency which will be explained in the numerical experiments.
The paper is organized as follows. Through the two equivalent forms of the Fornberg-Whitham type equations, we construct two dissipative, conservative DG schemes in Section 2 and 3, respectively. We demonstrate the dissipation and conservation correspondingly. Additionally, some results of the error estimate are stated. Subsequently, several numerical experiments are presented in Section 4 to show the capability of the methods. This paper is concluded in Section 5. Some more technical proofs of relevant lemmas are listed in the appendix.

Notations
We denote the mesh T h by with the cell center denoted by The finite element space as the solution and test function space consists of piecewise polynomials where P k (I j ) denotes the set of polynomials of degree up to k defined on the cell I j .
Notably, the functions in V k h are allowed to be discontinuous across cell interfaces. The values of u at x j+ , from the left cell I j and the right cell I j+1 , respectively. Additionally, the jump of u is defined as To simplify expressions, we adopt the round bracket and angle bracket for the L 2 inner product and boundary term on cell I j for the one dimensional case.

The LDG scheme
We consider the Fornberg-Whitham type equations where f (u) = 1 p u p , and p ≥ 2 is an integer. The periodic boundary condition is adopted here, which is not essential. The LDG methods we propose here or later can be designed for non-periodic boundary condition easily.
First, we split the equation (2.2) into a first-order system Then the semi-discrete LDG scheme is formulated as: Find numerical solutions for any test function φ, ϕ, ψ ∈ V k h . Here, the "hat" terms in (2.3) 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. The main distinctions between dissipative and conservative schemes are the choices of numerical fluxes. We have two LDG methods as follows: Scheme D1 : For the dissipative numerical flux of the nonlinear term f (u) here, we take the Godunov flux to prepare the optimal convergence rate subsequently, which is called an upwind numerical flux satisfying the following property, . (2.5) For the numerical fluxes q h , v h , u h , we select to maintain the energy stability. Numerically, the dissipative scheme (2.3) with fluxes (2.4), (2.6) can achieve (k + 1)-th order of accuracy.
Scheme C1 : For the conservative scheme C1, the numerical flux for the nonlinear term f (u) is given by as in [3]. Then we choose the central fluxes for q h , v h , u h , For the conservative scheme (2.3) with fluxes (2.7), (2.8) on uniform meshes, we obtain the k-th order for odd k, and (k + 1)-th order for even k numerically.

Dissipation and conservation
In this section, we provide proof of energy dissipation or energy conservation for the proposed LDG schemes in Section 2.2. The proposition demonstrates the conservative quantities based on which we construct the proposed numerical schemes D1 and C1, including mass E 0 , and energy E 2 in (1.3).
Before the proposition, we define some bilinear operators to simplify our expressions.
where the direction of w determines the operator L + , L − or L c . Wherein L ± denotes the cases with ω ± correspondingly, and L c is for ω = ω c .
The distinction between the dissipative form N d j and the conservative form N c j lies in the numerical flux f , which are taken in (2.4) or (2.7), respectively.
N c,d j , there hold the following properties, The properties in Lemma 2.1 can be easily derived by algebraic manipulation which had already been proved in [3], [27], so we do not give the details here.
Next, we prove the L 2 stability by taking the test functions as After summing up corresponding equations over all intervals, • For the dissipative scheme D1 • For the conservative scheme C1 Here we have used the results in Lemma 2.1. Then we obtain the final dissipation or conservation in (2.12) and (2.13).

Error estimates
In this section, we provide error estimates of the LDG schemes in Section 2.2 for the sufficiently smooth periodic solution of the Fornberg-Whitham type equations.

Notations, projections, and auxiliary results
First, we make some conventions for different constants. For any time t in [0, T ], we assume that the exact solution and its spatial derivatives are all bounded. We use the notation C to denote a positive constant which is independent of h, but depends on |f | and the exact solution of the problem considered in this paper. Additionally, the notation C * is used to denote the constants which are relevant to the maximum of |f |.
Under different circumstances, these constants will have different values.
Next, we will introduce some projection properties to be used later. The standard L 2 projection of a function ζ with k + 1 continuous derivatives into space V k h , is denoted by P, i.e., for each I j (Pζ − ζ, φ) I j = 0, ∀φ ∈ P k (I j ), and the Gauss Radau projections P ± into V k h satisfy, for each I j , ).
For the projections mentioned above, it is easy to show [4] that where ζ e = ζ − Pζ or ζ − P ± ζ, and the positive constant C only depends on ζ.
Then some inverse inequalities of the finite element space V k h will be applied in the subsequent proofs.

Lemma 2.3. [4]
For ∀ω ∈ V k h , there exists a positive constant C which is independent on ω, h, such that where the integer p is in the nonlinear term f (u) = 1 p u p . The constant C depends on the final time T , k, u k+2 and the bounds of derivatives up to second order of the nonlinear term f (u). Here, u k+2 is the maximum of the standard Sobolev k + 2 norm over [0, T ].

The error equation
Since the exact solution also satisfies the numerical scheme (2.3), doing subtraction can bring us the error equations. Due to the different choices of test functions in the error equations, we define the bilinear form B j as and the form for the nonlinear term f (u), which is linear with respect to its second argument, is Here, the notations ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ), ψ = (ψ 1 , ψ 2 , ψ 3 ) are vectors consisting of test functions in the finite element space V k h . We also define for writing convenience. After applying summation over all cells I j , the error equation Here, for the dissipative scheme D1, we define Without causing misunderstanding, the notations in (2.19) are still used to represent the analogs for the conservative scheme. For conservative scheme C1, we replace the Gauss-Radau projections by the standard L 2 projections Taking test functions we have the energy equality as H j (f ; u, u h , ξ u ).

The proof of the main results in Theorem 2.4
Next, we analyze B j and H j , respectively. Some primary estimate results will be provided in the following lemmas.
• Estimates for the linear terms Lemma 2.5. The following energy equality holds, Proof. The proof is provided in Appendix A.1.
Lemma 2.6. For the bilinear form B j , the following equations hold by projection prop- Proof. The proof is provided in Appendix A.2.
• Estimates for the nonlinear terms Subsequently, our attention is turned to H j involving the nonlinear term f (u), (2.23) The index "ref " denotes the direction of the value on each element interface depending on the flow direction of the exact solution u in the adjacent elements. For the dissipative scheme D1, For the conservative scheme C1, the direction of the boundary value is considered as for sufficient small h and k ≥ 1, where µ = 1 for even p, and µ = 2 for odd p.
Proof. The proof is provided in Appendix A.3.
for sufficient small h and k ≥ 2.
Proof. The proof is provided in Appendix A.4.
Proof. The proof is provided in Appendix A.5.
• Final error estimates in Theorem 2.4 Finally, we are ready to get the error estimates in Theorem 2.4. We divide the final error u − u h L 2 (I) into two parts: the projection error η u and the approximation error ξ u . Once we prove the order of the approximation errors, the results of Theorem 2.4 is derived by triangle inequality and the interpolation inequality (2.14) straightforwardly.
Therefore, we mainly focus on ξ u L 2 (I) .
Estimates in (2.16): Gathering together the estimates (2.21), (2.25), the final error estimate for Scheme D1 is listed as follows, where µ = 1 for even p, and µ = 2 for odd p.
Estimates in (2.17): Lemma 2.6 and Lemma 2.8 lead us to the error estimate for Scheme C1 Using the Young's inequality with weights as and inverse inequality (ii) in (2.15), we get There is a further result for the conservative scheme with even k and odd N , i.e. the optimal order of accuracy can be proved in [16].

The LDG scheme for equation (1.2)
Based on the form we construct another two LDG schemes, including a dissipative scheme and a conservative scheme named by Scheme D2, and Scheme C2, respectively. The conservation or dissipation and the corresponding error estimate are also provided in this section.

The LDG scheme
Referring to [29,31], we split the above equation (3.1) into with periodic boundary condition. Then we first rewrite the above equation (3.2) into a first-order system for all test functions φ, ψ ∈ V k h . For (3.3), we can also rewrite it into a first-order system: Subsequently, we define the LDG scheme for (3.6) as: Find numerical solutions w h , u h , Numerically, Scheme D2 can obtain the optimal order of accuracy for the variable u.
Scheme C2 : The conservative numerical flux for the nonlinear term f (u) is the same as (2.7), and then we choose the central fluxes for u h , r h , s h , p h , Numerically, Scheme C2 can obtain the k-th order when piecewise polynomials of odd degree k are used, and (k + 1)-th order for even degree k.

Dissipation and conservation
In this section, a proposition demonstrating dissipation or conservation for the proposed LDG schemes in Section 3.1 is stated, including E 1 conservation and energy E 2 dissipation or conservation.
Proposition 3.1. For periodic problems, we have Proof. Owing to the form of conservation law we give in equation (3.3), we can have the E 1 conservativeness with periodic boundary conditions for Scheme D2, C2 trivially.
Subsequently, we will begin the proof of E 2 dissipation or conservation. First, for the two equations in (3.5), we take a first-order temporal derivative as Since the numerical schemes (3.12), (3.7) hold for any test function in test space V k h , we choose After summation of corresponding equalities over all intervals, Lemma 2.1 leads us to the energy stability • For the conservative scheme C2 i.e. the results in (3.10) and (3.11).

Error estimates
In this section, we will provide error estimates of the LDG schemes in Section 3.  = (a, b), and the finite element space V k h , there hold the following error estimates for small enough h, where the integer p is in the nonlinear term f (u) = 1 p u p . The constant C depends on the final time T , k, u k+2 and the bounds of derivatives up to second order of the nonlinear term f (u). Here, u k+2 is the maximum of the standard Sobolev k + 2 norm over [0, T ].

The error equation
Combining the error equations with different test functions, we define the bilinear form After applying summation over all cells I j , the error equation is expressed by In the same way as Section 2.4.3, we define we have the energy equality as

The proof of the main results in Theorem 3.2
The relevant estimates for H j have already been stated in Section 2.4.4. We just give the estimate results forB j .
• Estimates for the linear terms Lemma 3.3. The following energy equality holds, Proof. The proof is provided in Appendix A.6.
Lemma 3.4. For the bilinear formsB j , the following equations hold by projection prop- Proof. The proof is provided in Appendix A.7.
• Final error estimates in Theorem 3.2 After the above preparations, the proof of Theorem 3.2 can be provided. We still pay our attention to the approximation error ξ u . After the triangle inequality and interpolation property (2.14), we complete the proof for Theorem 3.2.
Estimates in (3.13): Combining (3.16) and (2.25), the error estimate for the dissipative scheme D2 is given by, where µ = 1 for even p, and µ = 2 for odd p.
Estimates in (3.14): Together with the estimates (3.17) and (2.26), we have the following inequality for the conservative scheme C2, The Young's inequality and inverse inequality (ii) in (2.15) imply

Numerical experiments
In this section, we present several numerical experiments to illustrate the capability of our numerical schemes, including the errors, convergence rate tables, and some plots of numerical solutions. For the time discretization, we adopt the explicit third-order methods in [20,9]. Under the situation of degree k (k ≥ 2) polynomial approximation space, we take the time step as ∆t = α∆x (k+1)/3 , α = 0.1 to unify orders of temporal and spatial discretization. Without a specific explanation, the periodic boundary condition and uniform meshes are used. All examples were performed on a Windows desktop system using an Intel Core i5 processor, and programmed in Intel Visual Fortran.  where the computational domain is [0, 1]. However, there is no exact solution for these two initial datums. Compared with the results in [13], our dissipative scheme D1 and  D2 with a TVB limiter [5] can capture the shock without oscillation, see    Using the initial condition of the KdV equation in [2] as u(x, t) = 12 κ 2 1 e θ 1 + κ 2 2 e θ 2 + 2(κ 2 − κ 1 ) 2 e θ 1 +θ 2 + a 2 (κ 2 2 e θ 1 + κ 2 1 e θ 2 )e θ 1 +θ 2 (1 + e θ 1 + e θ 2 + a 2 e θ 1 +θ 2 ) 2 (4.4) where we model the interaction of two solitons in Figure 4 The numerical solutions in Figure 4.3 is consistent with the results in [16].

Example 4.4. Single peakon solution
Our numerical schemes also work for the peakon solution whose first derivative is finite discontinuous. The exact solution for the case p = 2 is

Example 4.5. Periodic peakon solutions
For the Fornberg-Whitham equation i.e. p = 2, we try to get the approximations for periodic peakon solutions [13] with period 2T p ,   where d ± = 1 6 (4 ± 3 4g + 4s − 2s 2 ), The parameters s, g are constants where s denotes the speed of the wave, and g concerns the shape of the wave. We give three different cases in Figure 4.5. When we take the speed s = 2, the peakon solution will tend to the cuspon solution, as the parameter g → 4 9 . Our proposed schemes have accurate numerical solutions for the different values of g. For a long time approximation, we use s = 2, g = 0.3 in solution (4.6) as an example to illustrate the differences among the four numerical schemes. In Figure 4.6, we take the degree of piecewise polynomial space k = 2. It can be seen that the dissipative schemes D1 and D2 become inaccurate due to the error of shape and decay of amplitude over a long temporal interval. The conservative schemes have more accurate approximation results than the dissipative ones. In Figure 4.7, we show the results when the degree k = 4. It tells that high order discretization methods can reduce shape error of waves effectively. The conservative property we prove is a semi-discrete property for our schemes, which implies the fully discretization has energy fluctuation, see Table 4.4.
However, the conservativeness can reduce the dissipation of energy so that the conservative schemes have better approximation over a long temporal interval. The shape error caused by dispersion error and phase speed error can be reduced by conservativeness or higher order accuracy, the detailed analysis can be accomplished by Fourier expansion and error dynamics [36,18].
In Table 4.3, we make a comparison of the CPU time for the four proposed LDG schemes. It is indicated that the dissipative scheme costs less time than the conservative scheme owing to the minimal stencils we choose, similar to [39]. On the other hand, Schemes D1, C1 in Section 2 are more effective than Schemes D2, C2 in Section 3, correspondingly.
x u0, u The LDG scheme for (1.2) can be applied more widely. More details will be given in our future work which is out of the scope of this paper.

Conclusion
In this paper, for the two different forms of the Fornberg-Whitham type equations, we construct dissipative LDG schemes D1 and D2, and conservative DG schemes C1 and C2.
For the dissipative schemes, the optimal order of accuracy can be achieved numerically, which are identical to the theoretical proof for odd p.
After summing up (2.3a) and (2.3c) (twice with different test functions) over all intervals, we obtain for the dissipative scheme D1, then Lemma 2.1 can be used to derive the equality (2.20). Notably, we can replace all operators L + , L − with L c in the above equation to get the same equality for the conservative scheme C1.

A.2 Proof of Lemma 2.6
There are two terms for B j , • For (2.21) of the dissipative scheme D1: Via the proof of L 2 dissipation and Lemma 2.5, we get As for the second term of (A.2.1), it can be derived by the properties of Gauss-Radau projection which causes the boundary terms to vanish, as well as the integral terms with spatial derivatives.
• For (2.22) of the conservative scheme C1: The first term of B j is obtained by the L 2 conservation and Lemma 2.5, i.e.
The second term of B j is derived by the property of L 2 projection, orthogonality to all polynomials of degree up to k. Thus the remaining is some boundary terms as we state in (2.22).
Therefore, we have the results (2.21) and (2.22) for the bilinear term B j .

A.3 Proof of Lemma 2.7
For the part T 1 , we use Taylor expansion on where f u ,f u are the mean values. Substituting the above equations into the equation (2.23), we obtain For writing convenience, we omit the subscripts j + 1 2 for the boundary terms, I j for the integral terms. And we need to explain the notations we have used in the above Notably, the value of ξ, η varies with the direction "ref ". This setting is for the following estimates. • There is an extra h by |f (u)| ≤ C * h, thus the estimate for the O 1 term can be written as Owing to the property of projection, notice that (f (u j )η u , ξ u x ) I j = 0, this is the reason for the first inequality. Because of the estimate f (u) − f (u j ) = O(h), we obtain the second inequality. And then the inverse inequalities (2.15) are applied for deriving the final result.
• O 2 term: After a simple integration by parts, the O 2 term becomes The first term O 1 2 can be easily controlled by C * ξ u 2 L 2 (I) . We mainly focus on the second term O 2 2 . Due to the classification of the sign of f (u), it needs to be discussed separately. For the case f (u) > 0 on I j ∪ I j+1 , we have the negative term in O 2 2 , which can be ignored. For the case f (u) ≤ 0 on I j ∪ I j+1 , the second term O 2 2 is simplified as Because of the negative term f (u)([[ξ u ]] + (ξ + − ξ u,+ )) 2 , we can omit it directly.
For the last case, i.e. f (u) change its sign, the term f (u) can provide an extra h, and then the situation becomes Hence we conclude that for f (u) with all non-negative derivative f (u), there is However, if there exist some points satisfying f (u) < 0, then we have Therefore, for the specific form f (u) = 1 p u p , the estimate for term O 2 is finished.
• O 3 term: With the definition e = u − u h , where the last inequality requires small enough h and k ≥ 1.
Subsequently, we divide the term where u * h depends on the sign of f (·) between (u + h ) j+ 1 where ζ is between (u + h ) j+ We have now accomplished the proof for Lemma 2.7.
Remark A.1. In [37,38], the authors have proved the optimal orders of accuracy for upwind fluxes in hyperbolic conservation laws. The differences between our cases and theirs are directions of the test functions, which cause suboptimal order of accuracy for even p in the above term O 2 .
Remark A.2. In [3], the authors have given the result of the k-th order of accuracy for the nonlinear part f (u) = 1 p u p , p ≥ 2. Here in our proof, we draw the same conclusion by another way to verify the above result.

A.5 Proof of Lemma 2.9
Referring to the technique in [16], integration of (2.27) gives v(t, ·) 2 L 2 (I) ≤ CG(t) (A.7.1) where Since the following relation holds, we obtain a significant estimate after integrating the above inequality

dζ.
Note that the derivatives Q (y) is positive and bounded for all y ≥ 1, k ≥ 3−μ 2 . There must existsC such that Q(C) = CT.
By the monotonicity of Q, we conclude that G ≤CG(0). With the aid of (A.7.1), the result of Lemma 2.9 is obtained.

A.6 Proof of Lemma 3.3
We take test functions in equations (3.7) and (3.12) as ϕ = φ = (u h ) t and s h , ψ = ϑ = (r h ) t and p h .
Owing to the L 2 projection, all integral terms in the second term ofB j vanish and we are left with the remaining boundary terms.