An Ultra-weak Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for the KdV–Burgers–Kuramoto Equation

Abstract

In this paper, we study an ultra-weak local discontinuous Galerkin (UWLDG) method for the KdV–Burgers–Kuramoto (KBK) type equation. While the standard UWLDG method is a powerful tool for efficiently solving high order equations, it faces challenges when applied to equations involving multiple spatial derivatives. We adopt a novel approach to discretize lower order spatial derivatives, enhancing the versatility of the UWLDG method. Additionally, we adopt generalized numerical fluxes to enhance the flexibility and extendibility of the UWLDG scheme. We introduce a class of global projections with multiple parameters to analyze the properties of these generalized numerical fluxes. With the aid of the special discretization approach and the global projections, we establish both stability and optimal error estimates of proposed method. The validity of our theoretical findings is demonstrated through numerical experiments.

A Some Detailed Proofs for the Global Projections

1.1 A.1 Determinant of $\mathcal {A}$

$$\begin{aligned} \det \left( \mathcal {A}\right) = \prod _{j=1}^{N} \det \left( A+\omega _N^jB\right) , \quad \omega _N=e^{\frac{2\pi \text {i}}{N}}, \end{aligned}$$

where A, B are $2\times 2$ matrices, $\mathcal {A}=\textrm{circ}(A,B,\textbf{0},\cdots ,\textbf{0})$.

Proof

We first introduce the following matrix

$$\begin{aligned} \mathcal {W}=\left( \begin{array}{ccccc} 1 &{} 1 &{} 1 &{} \cdots &{} 1 \\ 1 &{} \omega _N &{} \omega _N^2 &{} \cdots &{} \omega _N^{N-1} \\ 1 &{} \omega _N^2 &{} \omega _N^4 &{} \cdots &{} \omega _N^{2(N-1)} \\ \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ 1 &{} \omega _N^{N-1} &{} \omega _N^{2(N-1)} &{} \cdots &{} \omega _N^{(N-1)(N-1)} \end{array} \right) \otimes \textbf{I}_2, \end{aligned}$$

which is a quasi-Vandermonde matrix, it is non-singular for it is the Kronecker product of two invertible matrices. Then we can check the equality

$$\begin{aligned} \mathcal {A}\mathcal {W}=\mathcal {W}\varLambda , \end{aligned}$$

where $\varLambda =\text {diag}(A+B,A+\omega _NB,\cdots ,A+\omega _N^{N-1}B)$ is a quasi-diagonal matrix. Thus $\det \left( \mathcal {A}\right) = \det \left( \varLambda \right) = \prod _{j=1}^{N} \det \left( A+\omega _N^jB\right)$.$\square$

1.2 A.2 Invertibility of $A+\omega B$

Let $k\ge 1$, $\omega \in \mathbb {C}$ with $|\omega |=1$, and $(\theta ,\mu ) \in \mathbb {A}_2(k)$, then $\det \left( A+\omega B\right) \ne 0$, where A and B are defined in (3.10).

Proof

We can trivially compute that

$$\begin{aligned} \det \left( A+\omega B\right) = 2\left( \tilde{\theta }\tilde{\mu }\omega ^2 + (-1)^{k-1}k(\theta \tilde{\mu }+\tilde{\theta }\mu )\omega + \theta \mu \right) . \end{aligned}$$

Denote $P(\omega )=\frac{1}{2}\det \left( A+\omega B\right)$, due to the discriminant of $P(\omega )$ is

$$\begin{aligned} \varDelta = k^2(\theta \tilde{\mu }+\tilde{\theta }\mu )^2 -4\theta \mu \tilde{\theta }\tilde{\mu }\ge 0, \end{aligned}$$

then the roots of $P(\omega )$ must be real. Thus for $\omega$ that is not real, $P(\omega )\ne 0$. With noticing the values of $P(\pm 1)$ are

$$\begin{aligned} P(1) = \left( (-1)^{k-1}k-1\right) (\theta +\mu -2\theta \mu )+1&, \\ P(-1)= \left( (-1)^{k}k-1\right) (\theta +\mu -2\theta \mu )+1&, \end{aligned}$$

then we have $\det \left( A\pm B\right) =2P(\pm 1)\ne 0$ for $(\theta ,\mu )\in \mathbb {A}_2(k)$, which means $A+\omega B$ is invertible for all $|\omega |=1$ and $(\theta ,\mu )\in \mathbb {A}_2(k)$.