The Exponential Invariant Energy Quadratization Approach for General Multi-Symplectic Hamiltonian PDEs.

Many conservative systems in physical sciences can be described by multi-symplectic Hamiltonian PDEs (MS-HPDEs) admitting three important inherent properties named as multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we develop a novel strategy to systematically derive linearly implicit local energy-preserving schemes for general MS-HPDEs, which is named the exponential invariant energy quadratization approach (EIEQ). Such novel strategy is based on the exponential form of non-quadratic terms of the state function that can remove the bounded-from-below restriction, so it is more applicable than the traditional IEQ approach widely employed to construct linearly implicit schemes. Moreover, we provide a completely explicit discretization of the auxiliary variable combined with the nonlinear term, which obtains linearly implicit schemes of the constant coefficient, making their more effective than the IEQ schemes for rapid simulations. We also present the multiple EIEQ approach to improve the applicability of the EIEQ approach for MS-HPDEs with more non-quadratic terms. In addition, when periodic or homogeneous boundary conditions are considered, the proposed schemes are global energy-preserving and can be explicitly solved by using the fast Fourier transform. Finally, ample numerical tests are carried out to demonstrate the computational efficiency, conservation and accuracy of EIEQ schemes.