Journal
JOURNAL OF SCIENTIFIC COMPUTING
Volume 97, Issue 3, Pages -Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-023-02388-y
Keywords
Conservative scheme; High-order accuracy; Quadratic auxiliary variable approach; Symplectic Runge-Kutta method; Klein-Gordon-Schrodinger equations.
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We present a class of arbitrarily high-order conservative schemes for the Klein-Gordon Schrodinger equations, which combine the symplectic Runge-Kutta method with the quadratic auxiliary variable approach and can effectively preserve the conservation of energy and mass.
We present a class of arbitrarily high-order conservative schemes for the Klein-Gordon Schrodinger equations. These schemes combine the symplectic Runge-Kutta method with the quadratic auxiliary variable approach. We first introduce an auxiliary variable that satisfies a quadratic equation to reformulate the original system into an equivalent one. This reformulated system possesses two strong quadratic invariants: energy and mass. Next, we discretize the reformulated system using symplectic Runge-Kutta methods, yielding a class of semi-discrete systems with arbitrarily high-order accuracy in time. The semi-discrete systems naturally preserve the discrete contour part of the strong invariants and the relationship of the quadratic equation. By eliminating the intermediate variable, we obtain the original energy conservation law. Then, the Fourier pseudo-spectral method is employed to obtain the fully discrete scheme that preserves the original energy and mass. We provide a fast solver to implement the proposed methods effectively. Numerical experiments demonstrate the expected accuracy and conservation of proposed schemes.
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