Local Discontinuous Galerkin Methods for the \(\mu \)-Camassa–Holm and \(\mu \)-Degasperis–Procesi Equations

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the \(\mu \)-Camassa–Holm (\(\mu \)CH) and \(\mu \)-Degasperis–Procesi (\(\mu \)DP) equations. The conservative schemes for both two equations can preserve discrete versions of their own first two Hamiltonian invariants, while the dissipative ones guarantee the corresponding stability. The error estimates of both LDG schemes for the \(\mu \)CH equation are given. Comparing with the error estimates for the Camassa–Holm equation, some important tools are used to handle the unexpected terms caused by its particular Hamiltonian invariants. Moreover, a priori error estimates of two LDG schemes for the \(\mu \)DP equation are also proven in detail. Numerical experiments for both equations in different circumstances are provided to illustrate the accuracy and capability of these schemes and give some comparisons about their performance on simulations.

Authors and Affiliations

1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

   Chao Zhang, Yan Xu & Yinhua Xia

Corresponding author

Correspondence to Yinhua Xia.

Publisher's Note

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Y. Xu: Research supported by NSFC Grant Nos. 11722112, 91630207. Y. Xia: Research supported by NSFC Grant Nos. 11471306, 11871449, and a Grant from the Science and Technology on Reliability and Environmental Engineering Laboratory (No. 6142A0502020817).

Appendix

1.1 Proof of the Equivalence of (1.2) and (1.11)

Proof

Notice that

and

due to the periodicity, then (1.2) can be rewritten as

Apply the invertible operator \(A^{-1}_{\mu }\) to above equation and take into consideration of \(\mu (u)\) being a conservative quantity, then we have

Comparing the above equation and our desired (1.11), it remains to check the following claim:

Set \(v=A^{-1}_{\mu }u\), in other word, \(u=A_{\mu }v=\mu (v)-v_{xx}\). v is obviously periodic like u and we know that \(\mu (v)=\mu (u)\), which means that \(\mu (v)\) is also a conservative quantity. On account of the periodicity and conservation, the mean value \(\mu (v)=\int _{0}^{1} v dx\) is a constant no matter where and when, so we have \(\mu (v)_x=0\); furthermore, owing to the periodicity, we also get \(\mu (v_x)=0\).

Now we apply \(A_{\mu }\) to \(A^{-1}_{\mu }(u_x)-(A^{-1}_{\mu }u)_x\), then

Thus we have proven the claim, and further completed the equivalence of (1.11) and (1.2). \(\square \)

1.2 Proof of Lemma 3.7

Proof

By the Proposition 3.2, we know

Then by the orthogonality of \(L^2\) projection,

so

Then we can get \(\mu (s^e)=\mu ({\mathcal {P}}^{+}u)-\mu (u)=0\). In addition to the fact \(\mu (u)_t=0\), we have

Taking consideration of the definitions of s and \(s^e\), we easily get

\(\square \)

1.3 Proof of Lemma 3.8

Proof

By the same argument as that used for the stability in Proposition 3.2 and on account of the results in Lemma 3.7, the first term of the right-hand side in (A.1) becomes

where \(\Psi =v^{-}s^{-} - {\widehat{v}}s^{-} -\breve{s}v^{-}+\delta _t^{-}s^{-} - {\widehat{\delta }}_ts^{-} - {\widehat{s}}\delta _t^{-}\).

As to the second term of the right-hand side in (A.1), we have

where \(\Phi ={\widehat{v}}^{e}s^{-}-\breve{s}^{e}v^{-}-{\widehat{\delta }}_{t}^{e}s^{-}-{\widehat{s}}^{e}\delta _{t}^{-}\). Because \({\mathcal {P}}\) is a local \(L^2\) projection, and \({\mathcal {P}}^{+}\), although not a local \(L^2\) projection, does have the property that \(s-{\mathcal {P}}^{+}s\) is locally orthogonal to all polynomials of degree up to \(k-1\), we have

Noticing the special interpolation property of the projection \({\mathcal {P}}^{+}\), we have

Then equation (A.2) becomes

Combining the above equation with (A.1), summing over j, taking into account the periodic boundary condition, we obtain the desired equality (3.47). \(\square \)

1.4 Proof of Lemma 3.9

For the proof of this lemma, we follow the idea of Lemma 3.4 and 3.5 in [19]. For \(f(u)=2\mu (u)u\) in the \(\mu \)CH equation (1.1), we have \(f''(u)=0\) and \(f'''(u)=0\), then we could simplify the proof.

Proof

Review the equality (3.48), and the estimates for every part in the right-hand side of (3.48) is as following

• Conservative scheme

For the last term, if \({\widehat{f}}\) is chosen as (3.14), we get trivially

for the reason that \({\widehat{f}}=\frac{1}{2}\left( f(u_h^+)+f(u_h^-)\right) =2\mu _0(u_h^- + u_h^+)=f(\{u_h\})\).

• Dissipative scheme

Besides, when \({\widehat{f}}\) is chosen as the Lax-Friedrichs flux (3.15), via the fact \([u_h]=[u_h-u]=[s^e-s]\), we have

where \(\alpha =\max \nolimits _{u_h}|f'(u_h)|\ge 0\).

For the other two terms \(\sum _{j=1}^{N}\int _{I_j}(f(u)-f(u_h))s_xdx + \sum _{j=1}^{N}\left( f(u_h)-f((\{u_h\}))[s]\right) _{j+\frac{1}{2}}\), observing that \(f(u)=2\mu (u)u\) where \(\mu (u)\) is conservative, denoting \(C_{\mu }=2\mu (u)\), we can obtain

then

where

Making further analysis, we can get

Combining them and we can get the conclusion of Lemma 3.9

\(\square \)

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