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

Abstract

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.

Appendix

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

Proof

Notice that

$$\begin{aligned} (A_{\mu }u)_t=\mu (u)_t-u_{xxt} \end{aligned}$$

and

$$\begin{aligned} A_{\mu }\left( \left( \frac{1}{2}u^2\right) _x\right) =-\frac{1}{2}(u^2)_{xxx}=-3u_xu_{xx}-uu_{xxx} \end{aligned}$$

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

$$\begin{aligned} A_{\mu }(u_t) + A_{\mu }(uu_x) + 3\mu (u)u_x=0. \end{aligned}$$

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

$$\begin{aligned} u_t + uu_x + 3\mu (u)A^{-1}_{\mu }(u_x)=0. \end{aligned}$$

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

$$\begin{aligned} A^{-1}_{\mu }(u_x) = (A^{-1}_{\mu }u)_x. \end{aligned}$$

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

$$\begin{aligned}&A_{\mu }A^{-1}_{\mu }(u_x) - A_{\mu } (A^{-1}_{\mu }u)_x \nonumber \\&\quad = u_x - A_{\mu }(v_x) \nonumber \\&\quad = (A_{\mu }v)_x - A_{\mu }(v_x) \nonumber \\&\quad = (\mu (v)_x-(v_{xx})_x) - (\mu (v_x) -(v_x)_{xx}) \nonumber \\&\quad = \mu (v)_x - \mu (v_x) \nonumber \\&\quad = 0. \end{aligned}$$

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

$$\begin{aligned}&\mu (u_h)_t = 0. \end{aligned}$$

Then by the orthogonality of $L^2$ projection,

$$\begin{aligned}&s^e = ({\mathcal {P}}^{+}u-u)\perp {\mathbb {P}}^{k-1}, \end{aligned}$$

so

$$\begin{aligned} \int _{I}s^e \cdot 1 dx = 0. \end{aligned}$$

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

$$\begin{aligned}&\mu ({\mathcal {P}}^{+}u)_t=0. \end{aligned}$$

Taking consideration of the definitions of s and $s^e$, we easily get

$$\begin{aligned}&\frac{d}{dt}\mu (s) \equiv 0,\quad \frac{d}{dt}\mu (s^e)\equiv 0. \end{aligned}$$

$\square$

1.3 Proof of Lemma 3.8

Proof

$$\begin{aligned}&\sum _{j=1}^{N}{\mathcal {B}}_{j}(s-s^e,\xi -\xi ^e,v-v^e,\delta -\delta ^e;-s,\delta _t,s,-\delta ,v)\nonumber \\&\qquad =\sum _{j=1}^{N}{\mathcal {B}}_{j}(s,\xi ,v,\delta ;-s,\delta _t,s,-\delta ,v) - \sum _{j=1}^{N}{\mathcal {B}}_{j}(s^e,\xi ^e,v^e,\delta ^e;-s,\delta _t,s,-\delta ,v) \end{aligned}$$

(A.1)

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

$$\begin{aligned}&{\mathcal {B}}_j(s,\xi ,v,\delta ;-s,\delta _t,s,-\delta ,v)=\int _{I_j}(\delta _t\delta )dx+\Psi _{j+\frac{1}{2}}-\Psi _{j-\frac{1}{2}}, \end{aligned}$$

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

$$\begin{aligned}&{\mathcal {B}}_{j}(s^e,\xi ^e,v^e,\delta ^e;-s,\delta _t,s,-\delta ,v) \nonumber \\&\quad = \int _{I_j}\delta ^{e}\delta _{t}dx + \int _{I_j}(\delta ^{e}v-v^{e}\delta )dx +\int _{I_j}(\delta _{t}^{e}s_x+s^{e}(\delta _t)_{x}+v^{e}s_{x}+s^{e}v_{x})dx \nonumber \\&\qquad +\left( ({\widehat{v}}^{e}+{\widehat{\delta }}_{t}^{e})[s]\right) _{j-\frac{1}{2}}+\left( {\widehat{s}}^{e}[\delta _{t}]\right) _{j-\frac{1}{2}}+(\breve{s}^e[v])_{j-\frac{1}{2}}+\Phi _{j+\frac{1}{2}}-\Phi _{j-\frac{1}{2}}, \end{aligned}$$

(A.2)

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

$$\begin{aligned}&\int _{I_j}\delta ^{e}\delta _{t}dx + \int _{I_j}(\delta ^{e}v-v^{e}\delta )dx +\int _{I_j}(\delta _{t}^{e}s_x+s^{e}(\delta _t)_{x}+v^{e}s_{x}+s^{e}v_{x})dx =0. \end{aligned}$$

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

$$\begin{aligned}&({\widehat{s}}^{e}[\delta _{t}])_{j-\frac{1}{2}}+(\breve{s}^e[v])_{j-\frac{1}{2}}=0. \end{aligned}$$

Then equation (A.2) becomes

$$\begin{aligned}&{\mathcal {B}}_{j}(s^e,\xi ^e,v^e,\delta ^e;-s,\delta _t,s,-\delta ,v)= \left( ({\widehat{v}}^{e}+{\widehat{\delta }}_{t}^{e})[s]\right) _{j-\frac{1}{2}}+\Phi _{j+\frac{1}{2}}-\Phi _{j-\frac{1}{2}}. \end{aligned}$$

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

$$\begin{aligned}&\sum _{j=1}^{N}\left( \left( f(\{u_h\})-{\widehat{f}}\right) [s]\right) _{j+\frac{1}{2}} = 0, \end{aligned}$$

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

$$\begin{aligned}&\sum _{j=1}^{N}\left( \left( f(\{u_h\})-{\widehat{f}}\right) [s]\right) _{j+\frac{1}{2}} = \frac{\alpha }{2}\sum _{j=1}^{N}\left( [u_h][s] \right) _{j+\frac{1}{2}} \\&\quad =\frac{\alpha }{2}\sum _{j=1}^{N}\left( [s^e][s] \right) _{j+\frac{1}{2}}-\frac{\alpha }{2}\sum _{j=1}^{N}\left( [s][s] \right) _{j+\frac{1}{2}}\\&\quad \le -\frac{\alpha }{2}\sum _{j=1}^{N}\left( [s]^2 \right) _{j+\frac{1}{2}} + Ch^{2k+1}, \end{aligned}$$

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

$$\begin{aligned}&f(u)-f(u_h)=C_{\mu }(s-s^e),\\&f(u)-f(\{u_h\})=C_{\mu }(\{s\}-\{s^e\}), \end{aligned}$$

then

$$\begin{aligned}&\sum _{j=1}^{N}\int _{I_j}(f(u)-f(u_h))s_xdx + \sum _{j=1}^{N}\left( f(u_h)-f((\{u_h\}))[s]\right) _{j+\frac{1}{2}}\nonumber \\&\quad =\sum _{j=1}^{N}\int _{I_j}C_{\mu }ss_xdx+\sum _{j=1}^{N}C_{\mu }(\{s\}[s])_{j+\frac{1}{2}} -\left( \sum _{j=1}^{N}\int _{I_j}C_{\mu }s^es_xdx+\sum _{j=1}^{N}C_{\mu }(\{s^e\}[s])_{j+\frac{1}{2}}\right) \nonumber \\&\quad ={\mathscr {T}}_1+{\mathscr {T}}_2, \end{aligned}$$

where

$$\begin{aligned} {\mathscr {T}}_1&= \sum _{j=1}^{N}\int _{I_j}C_{\mu }ss_xdx+\sum _{j=1}^{N}C_{\mu }(\{s\}[s])_{j+\frac{1}{2}},\\ {\mathscr {T}}_2&= -\left( \sum _{j=1}^{N}\int _{I_j}C_{\mu }s^es_xdx+\sum _{j=1}^{N}C_{\mu }(\{s^e\}[s])_{j+\frac{1}{2}}\right) . \end{aligned}$$

Making further analysis, we can get

$$\begin{aligned} {\mathscr {T}}_1&= C_{\mu }\left( \sum _{j=1}^{N}\int _{I_j}\frac{1}{2}(s^2)_xdx + \sum _{j=1}^{N}\left( \frac{s^+ + s^-}{2}(s^+-s^-)\right) _{j+\frac{1}{2}}\right) =0, \\ {\mathscr {T}}_2&= -C_{\mu }\left( \sum _{j=1}^{N}\int _{I_j}s^es_xdx + \sum _{j=1}^{N}(\{s^e\}[s])_{j+\frac{1}{2}} \right) \nonumber \\&\le C_{\mu }\Vert s^e\Vert \Vert s_x\Vert + C_{\mu } \Vert s^e\Vert _{\Gamma _h} \Vert s\Vert _{\Gamma _h} \nonumber \\&\le C\Vert s\Vert ^2 + Ch^{2k}. \end{aligned}$$

Combining them and we can get the conclusion of Lemma 3.9

$$\begin{aligned}&\sum _{j=1}^{N}{\mathcal {H}}_{j}(f;u,u_h;s) \le C \Vert s\Vert ^{2}+Ch^{2k}. \end{aligned}$$

$\square$