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Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime

Authors: Yongyong Cai and Yongjun Yuan
Journal: Math. Comp. 87 (2018), 1191-1225
MSC (2010): Primary 35Q55, 65M06, 65M12, 65M15
Published electronically: October 12, 2017
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Abstract: We rigorously analyze the error estimates of the conservative finite difference method (CNFD) for the Zakharov system (ZS) with a dimensionless parameter $ \varepsilon \in (0,1]$, which is inversely proportional to the ion acoustic speed. When $ \varepsilon \to 0^+$, ZS collapses to the standard nonlinear Schrödinger equation (NLS). In the subsonic limit regime, i.e., $ \varepsilon \to 0^+$, there exist highly oscillatory initial layers in the solution. The initial layers propagate with $ O(\varepsilon )$ wavelength in time, $ O(1)$ and $ O(\varepsilon ^2)$ amplitudes, for the ill-prepared initial data and well-prepared initial data, respectively. This oscillatory behavior brings significant difficulties in analyzing the errors of numerical methods for solving the Zakharov system. In this work, we show the CNFD possesses the error bounds $ h^2/\varepsilon +\tau ^2/\varepsilon ^3$ in the energy norm for the ill-prepared initial data, where $ h$ is mesh size and $ \tau $ is time step. For the well-prepared initial data, CNFD is uniformly convergent for $ \varepsilon \in (0,1]$, with second-order accuracy in space and $ O(\tau ^{4/3})$ accuracy in time. The main tools involved in the analysis include cut-off technique, energy methods, $ \varepsilon $-dependent error estimates of the ZS, and $ \varepsilon $-dependent error bounds between the numerical approximate solution of the ZS and the solution of the limit NLS. Our approach works in one, two and three dimensions, and can be easily extended to the generalized Zakharov system and nonconservative schemes. Numerical results suggest that the error bounds are sharp for the plasma densities and the error bounds of the CNFD for the electric fields are the same as those of the splitting methods.

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Additional Information

Yongyong Cai
Affiliation: Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China – and – Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Yongjun Yuan
Affiliation: Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha 410006, People’s Republic of China – and – Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China

Keywords: Zakharov system, error estimates, subsonic limit, finite difference method, conservative scheme
Received by editor(s): September 28, 2015
Received by editor(s) in revised form: June 19, 2016, and December 23, 2016
Published electronically: October 12, 2017
Additional Notes: The first author was partially supported by NSF grants DMS-1217066 and DMS-1419053, and by the NSAF grant U1530401; the second and corresponding author was partially supported by the National Natural Science Foundation of China under grant 11601148, and by the Construct Program of the Key Discipline in Hunan Province.
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