Abstract
In this work, two novel classes of structure-preserving spectral Galerkin methods are proposed which based on the Crank–Nicolson scheme and the exponential scalar auxiliary variable method respectively, for solving the coupled fractional nonlinear Klein–Gordon–Schrödinger equation. The paper focuses on the theoretical analyses and computational efficiency of the proposed schemes, the Crank–Nicoloson scheme is proved to be unconditionally convergent and has maximum-norm boundness of numerical solutions. The exponential scalar auxiliary variable scheme is linearly implicit and decoupled, but lack of the maximum-norm boundness, also, the energy structure is modified. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Both the theoretical analyses and the numerical comparisons show that the proposed spectral Galerkin methods have high efficiency in long-time computations.
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Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 12171245, 11971242) and the Research Start-up Foundation of Jiangxi Normal University (Grant No. 12021997).
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Appendices
A Appendix: Proof of (3.11)
Proof
Utilizing the Taylor’s expansion for \(\phi \) and \(\psi \) of (3.5) at \(t=t_{n+1/2}\), we have
in which
and
Assume that \(\phi (\cdot ,t)\in C^3([0,T])\). We deduce
Simultaneously,
Employing the Lagrange’s mean value theorem, we obtain
and
Suppose that \(\phi (\cdot ,t)\in C^4([0,T])\). We get
This completes the proof of (3.11). \(\square \)
B Appendix: Proof of Theorem 3.3
The proof of Theorem 3.3 is divided into two parts, including the existence and uniqueness.
(I) Existence:
Proof
It is worth noting that \(U_N^{n+1}=2U_N^{n+1/2}-U_N^{n}\) and \(\delta _t\Phi _N^{n+1/2}=\Psi _N^{n+1/2}\). We reformulate the spectral Galerkin scheme (3.12)–(3.14) into the following form
Now, we carry out proving the existence of \(U_N^{n+1/2}\) and \(\Phi _N^{n+1/2}\). For convenience, we define the map \({\textbf {s}}=(s_1,s_2)\), \(\mathscr {F}=(\mathscr {F}_1,\mathscr {F}_2):\) \((X_N^0(\Omega ),X_N^0(\Omega ))\rightarrow (X_N^0(\Omega ),X_N^0(\Omega ))\), such that
and
Choosing \(w_N=s_1\) in (B.3) and taking the real part, from the Young’s inequality, we derive
Setting \(w_N=s_2\) in (B.4), by utilizing the Cauchy–Schwarz inequality, we obtain
By employing the boundness of the numerical solutions (see Theorem 3.2) and the Young’s inequality, for sufficient small \(\tau \), we conclude
and
Substituting (B.6)–(B.8) into (B.5), we arrive at
Therefore, we have
Taking \(\delta =\Big \Vert \Big ( 2\Phi _N^{n}+\tau \Psi _N^{n}, \sqrt{2\mathcal {C}}\Big ) \Big \Vert \), which satisfies the condition of Lemma 3.5, we derive
This proves the existence of the numerical solution of (3.12)–(3.14). \(\square \)
Next, we prove the uniqueness of the numerical solution.
(II) Uniqueness:
Proof
We prove the theorem by introduction. It is obvious to find from (3.15) that the numerical solution \((U_N^0,\Phi ^0_N,\Psi _N^0)\in (X_N^0(\Omega ),X_N^0(\Omega ),X_N^0(\Omega ))\) exists and is unique. Assume that \((U_N^n,\Phi ^n_N,\Psi _N^n)\) is the unique solution of (3.12)–(3.14) for \(n=0,1,\ldots ,N_t-1\). Next, we prove the uniqueness of the solution \((U_N^{n+1/2},\Phi ^{n+1/2}_N)\). Assume there are two solutions \(X^{n+1/2}=(X_1^{n+1/2},X_2^{n+1/2})\) and \(Y^{n+1/2}=(Y_1^{n+1/2},Y_2^{n+1/2})\) for scheme (3.12)–(3.14). Then \(X_1^{n+1/2}-Y_1^{n+1/2}\) and \(X_2^{n+1/2}-Y_2^{n+1/2}\) satisfy (B.3) and (B.4) as follows
By the definition of \(\mathscr {F}_1\), we have
Taking the real part of (B.9), by admitting the boundness of the numerical solutions (see Theorem 3.2), Lemma 3.4 and the Cauchy–Schwarz inequality, we obtain
Taking into account of the definition of \(\mathscr {F}_2\), we get
By employing Theorem 3.2, Lemma 3.4 and the Cauchy–Schwarz inequality, we deduce
Summing up (B.10) and (B.11), for a sufficient small \(\tau \), we arrive at
By assuming \(\mathcal {C}\tau <1\), it leads to \(\Big \Vert X^{n+1/2}-Y^{n+1/2} \Big \Vert =0\), which implies \(X_1^{n+1/2}=Y_1^{n+1/2}\) and \(X_2^{n+1/2}=Y_2^{n+1/2}\). This ends the proof of the uniqueness. \(\square \)
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Hu, D., Fu, Y., Cai, W. et al. Unconditional Convergence of Conservative Spectral Galerkin Methods for the Coupled Fractional Nonlinear Klein–Gordon–Schrödinger Equations. J Sci Comput 94, 70 (2023). https://doi.org/10.1007/s10915-023-02108-6
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DOI: https://doi.org/10.1007/s10915-023-02108-6
Keywords
- Riesz fractional derivative
- Spectral Galerkin method
- Structure-preserving algorithm
- Unique solvability
- Convergence