High-order conservative energy quadratization schemes for the Klein-Gordon-Schrödinger equation

Abstract

In this paper, we design two classes of high-accuracy conservative numerical algorithms for the nonlinear Klein-Gordon-Schrödinger system in two dimensions. By introducing the energy quadratization technique, we first transform the original system into an equivalent one, where the energy is modified as a quadratic form. The Gauss-type Runge-Kutta method and the Fourier pseudo-spectral method are then employed to discretize the reformulation system in time and space, respectively. The fully discrete schemes inherit the conservation of mass and modified energy and can reach high-order accuracy in both temporal and spatial directions. In order to complement the proposed schemes and speed up the calculation, we also develop another class of conservative schemes combined with the prediction-correction technique. Numerous experimental results are reported to demonstrate the efficiency and high accuracy of the new methods.

Appendix: A

We take 2-stage GRK method as an example to illustrate the process of Scheme 3.2 in detail. For the iterative values ${P_{i}^{n}}, {Q_{i}^{n}}$ and ${{\Phi }_{i}^{n}}\ (i=1,2)$, we first denote

$$ {X_{i}^{n}}:=\frac{({P_{i}^{n}})^{2}+({Q_{i}^{n}})^{2}}{\mathcal{K}^{n}},\quad {Y_{i}^{n}}:=\frac{{P_{i}^{n}}{{\Phi}_{i}^{n}}}{\mathcal{K}^{n}},\quad {Z_{i}^{n}}:=\frac{{Q_{i}^{n}}{{\Phi}_{i}^{n}}}{\mathcal{K}^{n}}. $$

Then, from the last three rows of (3.3), one has

$$ \begin{array}{@{}rcl@{}} \dot{K}_{i}^{n}&=&\frac{\langle\dot{\Phi}_{i}^{n},({P_{i}^{n}})^{2}+({Q_{i}^{n}})^{2}\rangle}{2\mathcal{K}^{n}} +\frac{\langle\dot{P}_{i}^{n},{P_{i}^{n}}{{\Phi}_{i}^{n}}\rangle}{\mathcal{K}^{n}} +\frac{\langle\dot{Q}_{i}^{n},{Q_{i}^{n}}{{\Phi}_{i}^{n}}\rangle}{\mathcal{K}^{n}}\\ &=&\frac{1}{2}\langle \dot{\Phi}_{i}^{n}, {X_{i}^{n}}\rangle +\langle \dot{P}_{i}^{n}, {Y_{i}^{n}}\rangle+\langle \dot{Q}_{i}^{n}, {Z_{i}^{n}}\rangle \triangleq \mathrm{I}_{i}^{n}+\text{II}_{i}^{n}+\text{III}_{i}^{n}, \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \dot{\Phi}_{i}^{n}&=&W^{n}+\tau\sum\limits_{j=1}^{2}a_{ij}\big[({\Delta}_{f}-\beta^{2}) {{\Phi}_{j}^{n}}+{K_{j}^{n}}{X_{j}^{n}}\big]\\ &=&W^{n}+\tau\sum\limits_{j=1}^{2}a_{ij}\big[({\Delta}_{f}-\beta^{2})({\Phi}^{n}+\tau \sum\limits_{\ell=1}^{2}a_{j\ell}\dot{\Phi}_{\ell}^{n})+(K^{n}+\tau\sum\limits_{\ell=1}^{2}a_{j\ell} \dot{K}_{\ell}^{n}){X_{j}^{n}}\big]. \end{array} $$

The above equality is rewritten explicitly and it yields

$$ \begin{array}{@{}rcl@{}} &\mathcal{A}\dot{\Phi}_{1}^{n}+\mathcal{B}\dot{\Phi}_{2}^{n}={{\Gamma}_{1}^{n}} +\tau^{2}(a_{11}^{2}{X_{1}^{n}}+a_{12}a_{21}{X_{2}^{n}})\dot{K}_{1}^{n} +\tau^{2}(a_{11}a_{12}{X_{1}^{n}}+a_{12}a_{22}{X_{2}^{n}})\dot{K}_{2}^{n},\\ &\mathcal{C}\dot{\Phi}_{1}^{n}+\mathcal{A}\dot{\Phi}_{2}^{n}={{\Gamma}_{2}^{n}} +\tau^{2}(a_{21}a_{11}{X_{1}^{n}}+a_{22}a_{21}{X_{2}^{n}})\dot{K}_{1}^{n} +\tau^{2}(a_{21}a_{12}{X_{1}^{n}}+a_{22}^{2}{X_{2}^{n}})\dot{K}_{2}^{n}, \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} \mathcal{A}&=&1-\tau^{2}(a_{11}^{2}+a_{12}a_{21})({\Delta}_{f}-\beta^{2}),\ \ \mathcal{B}=-2\tau^{2}a_{11}a_{12}({\Delta}_{f}-\beta^{2}),\ \ \mathcal{C}=-2\tau^{2}a_{11}a_{21}({\Delta}_{f}-\beta^{2}),\\ {{\Gamma}_{i}^{n}}&=&W^{n}+\tau(a_{i1}+a_{i2})({\Delta}_{f}-\beta^{2}){\Phi}^{n} +\tau(a_{i1}{X_{1}^{n}}+a_{i2}{X_{2}^{n}})K^{n}, \end{array} $$

and the coefficient property of 2-stage GRK method a₁₁ = a₂₂ was used. Some simple calculations imply that

$$ \dot{\Phi}_{i}^{n}={A_{i}^{n}}+{B_{i}^{n}}\dot{K}_{1}^{n}+{C_{i}^{n}}\dot{K}_{2}^{n}\quad \text{for $i=1,2$,} $$

where ${A_{i}^{n}}, {B_{i}^{n}}$ and ${C_{i}^{n}}$ are known. Similarly, from the other relations of (3.3), we get

$$ \dot{P}_{i}^{n}=\overline{A}_{i}^{n}+\overline{B}_{i}^{n}\dot{K}_{1}^{n} +\overline{C}_{i}^{n}\dot{K}_{2}^{n},\quad \dot{Q}_{i}^{n}=\widetilde{A}_{i}^{n}+\widetilde{B}_{i}^{n}\dot{K}_{1}^{n} +\widetilde{C}_{i}^{n}\dot{K}_{2}^{n}\quad \text{for} i=1,2. $$

Computing the discrete inner product of three relations mentioned above with ${X_{i}^{n}}, {Y_{i}^{n}}$ and ${Z_{i}^{n}}$, respectively, for i = 1, 2, one has

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{lllll} 2\mathrm{I}_{i}^{n}&=\langle\dot{\Phi}_{i}^{n},{X_{i}^{n}}\rangle =\langle {A_{i}^{n}},{X_{i}^{n}}\rangle +\langle {B_{i}^{n}},{X_{i}^{n}}\rangle (\mathrm{I}_{1}^{n}+\text{II}_{1}^{n}+\text{III}_{1}^{n}) +\langle {C_{i}^{n}},{X_{i}^{n}}\rangle (\mathrm{I}_{2}^{n}+\text{II}_{2}^{n}+\text{III}_{2}^{n}),\\ \text{II}_{i}^{n}&=\langle\dot{P}_{i}^{n},{Y_{i}^{n}}\rangle =\langle \overline{A}_{i}^{n},{Y_{i}^{n}}\rangle +\langle \overline{B}_{i}^{n},{Y_{i}^{n}}\rangle (\mathrm{I}_{1}^{n}+\text{II}_{1}^{n}+\text{III}_{1}^{n}) +\langle \overline{C}_{i}^{n},{Y_{i}^{n}}\rangle (\mathrm{I}_{2}^{n}+\text{II}_{2}^{n}+\text{III}_{2}^{n}),\\ \text{III}_{i}^{n}&=\langle\dot{Q}_{i}^{n},{Z_{i}^{n}}\rangle =\langle \widetilde{A}_{i}^{n},{Z_{i}^{n}}\rangle +\langle \widetilde{B}_{i}^{n},{Z_{i}^{n}}\rangle (\mathrm{I}_{1}^{n}+\text{II}_{1}^{n}+\text{III}_{1}^{n}) +\langle \widetilde{C}_{i}^{n},{Z_{i}^{n}}\rangle (\mathrm{I}_{2}^{n}+\text{II}_{2}^{n}+\text{III}_{2}^{n}). \end{array}\right. \end{array} $$

We solve the linear algebraic equations and obtain the values of $\mathrm {I}_{i}^{n}, \text {II}_{i}^{n}$ and $\text {III}_{i}^{n}\ (i=1,2)$. These results lead to the values of $\dot {K}_{i}^{n}$ and $\dot {P}_{i}^{n}, \dot {Q}_{i}^{n}, \dot {\Phi }_{i}^{n}$ sequentially. In terms of the iterative errors of ${P_{i}^{n}}, {Q_{i}^{n}}$ and ${{\Phi }_{i}^{n}}$, one could determine the values of $\dot {\Upsilon }_{i}^{n}\ ({\Upsilon }=P, Q, {\Phi }, W, K)$ and finally get the numerical solutions at the next time level from (3.4) and update the iterative values.