# American Institute of Mathematical Sciences

## Efficient and accurate sav schemes for the generalized Zakharov systems

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA 2 School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High-Performance Scientific Computing, Xiamen University, Xiamen 361005, China

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: This work is partially supported by NSF grant DMS-1720442 and NSFC grant 11971407

We develop in this paper efficient and accurate numerical schemes based on the scalar auxiliary variable (SAV) approach for the generalized Zakharov system and generalized vector Zakharov system. These schemes are second-order in time, linear, unconditionally stable, only require solving linear systems with constant coefficients at each time step, and preserve exactly a modified Hamiltonian. Ample numerical results are presented to demonstrate the accuracy and robustness of the schemes.

Citation: Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020262
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Numerical solutions of the electric field $|E(x,t)|^2$ at T = 1
Numerical solutions of the electric field $|E(x,t)|^2$ at $[0,1]$
Numerical and reference solutions for Case Ⅲ at T = 2 (left column) and T = 4 (right column)
Numerical solutions in (4.2): surface-plots of the electron density $|E(x,y,t)|^2$ (left) and ion density fluctuation $N(x,y,t)$ (right) for Case 1 with $\gamma = 0.8$ and $\lambda = 20$
Numerical solutions in (4.2): surface-plots of the electron density $|E(x,y,t)|^2$ (left) and ion density fluctuation $N(x,y,t)$ (left) with $\gamma = 0.1$ and $\lambda = 20$
Numerical solutions in (4.2): surface-plots of the electron density $|E(x,y,t)|^2$ (left) and ion density fluctuation $N(x,y,t)$ (left) for Case 1 with $\gamma = 0.1$ and $\lambda = 100$
Numerical solutions in (4.3) for Case Ⅰ with $\lambda = 2$
Numerical solutions in (4.3) for Case Ⅱ with $\lambda = 2$
Numerical solutions in (4.3) for Case Ⅲ with $\lambda = 2$
Numerical solutions in (4.3) for Case Ⅲ with $\lambda = 100$
Evolution of the total wave energy $\|\textbf{E}(t)\|^2$ and the wave energy of the three components of the electric field $\|E_1(t)\|^2$, $\|E_2(t)\|^2$ and $\|E_3(t)\|^2$ in (4.4) for Case 1 (left) and Case 2 (right)
Error and convergence rates in time
 $\delta t$ $|e_E|_{L^{\infty}(0,T;L^{\infty})}$ $Rate$ $|e_N|_{L^{\infty}(0,T;L^{\infty})}$ $Rate$ $2\times 10^{-2}$ 1.90E(-3) - 1.60E(-3) - $1\times 10^{-2}$ 4.82E(-4) 1.98 3.94E(-4) 1.98 $5\times 10^{-3}$ 1.21E(-4) 1.99 9.93E(-5) 1.99 $2.5\times 10^{-3}$ 3.04E(-5) 2.00 2.49E(-5) 2.00 $1.25\times 10^{-3}$ 7.61E(-6) 2.00 6.23E(-6) 2.00 $6.25\times 10^{-4}$ 1.90E(-6) 2.00 1.56E(-6) 2.00 $3.125\times 10^{-4}$ 4.76E(-7) 2.00 3.93E(-7) 1.99 $1.5625\times 10^{-4}$ 1.21E(-7) 1.97 1.08E(-7) 1.87
 $\delta t$ $|e_E|_{L^{\infty}(0,T;L^{\infty})}$ $Rate$ $|e_N|_{L^{\infty}(0,T;L^{\infty})}$ $Rate$ $2\times 10^{-2}$ 1.90E(-3) - 1.60E(-3) - $1\times 10^{-2}$ 4.82E(-4) 1.98 3.94E(-4) 1.98 $5\times 10^{-3}$ 1.21E(-4) 1.99 9.93E(-5) 1.99 $2.5\times 10^{-3}$ 3.04E(-5) 2.00 2.49E(-5) 2.00 $1.25\times 10^{-3}$ 7.61E(-6) 2.00 6.23E(-6) 2.00 $6.25\times 10^{-4}$ 1.90E(-6) 2.00 1.56E(-6) 2.00 $3.125\times 10^{-4}$ 4.76E(-7) 2.00 3.93E(-7) 1.99 $1.5625\times 10^{-4}$ 1.21E(-7) 1.97 1.08E(-7) 1.87
Discretization Error in space
 $N$ 32 64 128 256 $|e_E|_{L^{\infty}(0,T;L^{\infty})}$ 5.40E(-1) 7.84E(-2) 1.91E(-4) 8.88E(-7) $|e_N|_{L^{\infty}(0,T;L^{\infty})}$ 2.64E(-1) 1.23E(-1) 1.10E(-3) 5.19E(-6)
 $N$ 32 64 128 256 $|e_E|_{L^{\infty}(0,T;L^{\infty})}$ 5.40E(-1) 7.84E(-2) 1.91E(-4) 8.88E(-7) $|e_N|_{L^{\infty}(0,T;L^{\infty})}$ 2.64E(-1) 1.23E(-1) 1.10E(-3) 5.19E(-6)
Error and convergence rates for the conserved quantities
 $\delta t$ $|e_{D^{GZS}}|_{L^{\infty}(0,T)}$ $Rate$ $|e_P^{GZS}|_{L^{\infty}(0,T)}$ $Rate$ $|e_H^{GZS}|_{L^{\infty}(0,T)}$ $Rate$ $8\times 10^{-2}$ 1.50E(-3) - 2.10E(-3) - 2.00E(-3) - $4\times 10^{-2}$ 1.69E(-4) 3.14 4.68E(-4) 2.16 2.26E(-4) 3.10 $2\times 10^{-2}$ 2.05E(-5) 3.04 1.12E(-4) 2.07 2.90E(-5) 3.02 $1\times 10^{-2}$ 2.53E(-6) 3.02 2.73E(-5) 2.03 3.60E(-6) 3.01 $5\times 10^{-3}$ 3.14E(-7) 3.01 6.74E(-6) 2.02 4.49E(-7) 3.00 $2.5\times 10^{-3}$ 3.91E(-8) 3.01 1.68E(-6) 2.01 5.61E(-8) 3.00 $1.25\times 10^{-3}$ 4.88E(-9) 3.00 4.18E(-7) 2.00 7.09E(-9) 2.98 $6.25\times 10^{-4}$ 6.09E(-10) 3.00 1.05E(-7) 1.99 1.18E(-9) 2.58
 $\delta t$ $|e_{D^{GZS}}|_{L^{\infty}(0,T)}$ $Rate$ $|e_P^{GZS}|_{L^{\infty}(0,T)}$ $Rate$ $|e_H^{GZS}|_{L^{\infty}(0,T)}$ $Rate$ $8\times 10^{-2}$ 1.50E(-3) - 2.10E(-3) - 2.00E(-3) - $4\times 10^{-2}$ 1.69E(-4) 3.14 4.68E(-4) 2.16 2.26E(-4) 3.10 $2\times 10^{-2}$ 2.05E(-5) 3.04 1.12E(-4) 2.07 2.90E(-5) 3.02 $1\times 10^{-2}$ 2.53E(-6) 3.02 2.73E(-5) 2.03 3.60E(-6) 3.01 $5\times 10^{-3}$ 3.14E(-7) 3.01 6.74E(-6) 2.02 4.49E(-7) 3.00 $2.5\times 10^{-3}$ 3.91E(-8) 3.01 1.68E(-6) 2.01 5.61E(-8) 3.00 $1.25\times 10^{-3}$ 4.88E(-9) 3.00 4.18E(-7) 2.00 7.09E(-9) 2.98 $6.25\times 10^{-4}$ 6.09E(-10) 3.00 1.05E(-7) 1.99 1.18E(-9) 2.58
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