The development of highly accurate and structure-preserving numerical methods for solving partial differential equations is an ongoing quest even after decades of successful approaches. The research is particularly fuelled by recent demands arising from various applications in sciences and engineering. The significance of its numerical strategies has been universally acknowledged and validated through improvement of discrete methods in diverse branches, including finite difference methods, finite element methods, spectral collocation approaches, spectral Galerkin methods, and so on. In recent years, structure preserving methods, also known as geometric numerical integrators, have also emerged as a central topic in computational mathematics. It has been realized that an integrator should be designed to preserve as much as possible intrinsic features of the underlying problems, such as conservations of the mass, momentum and energy, as well as the symplecticity and multisymplecticity of Hamiltonian systems. Structure-preserving algorithms can be effectively utilized for simulations of a variety of theoretical and application problems, ranging from celestial mechanics, quantum mechanics, fluid dynamics, and geophysics.
This special issue is dedicated to recent advances in aforementioned pursuits for high-accuracy and structure-preserving algorithms when partial differential equations are targeted. We intend to accommodate a broad spectrum of investigations. Contributed papers in this special issue address, in particular, concerns in fields of:
mass conservation in least-squares finite element methods;
Fourier pseudo-spectral conservative schemes for Klein–Gordon–Schrödinger equation;
high-order IMEX-WENO finite volume approximations;
local discontinuous Galerkin methods for Hamiltonian partial differential equations;
structure-preserving exponential methods for Burgers–Huxley equation; and
energy-preserving schemes for 2D Hamiltonian wave equations with Neumann boundary conditions.
The numerical methods proposed by authors of this special issue have demonstrated that highly accurate and structure-preserving methods possess remarkable and reliable performances for solving different partial differential equations, especially in the long-time stability, which results from the preservation of intrinsic features of the underlying problems. In addition, the constructions of these powerful methods are very flexible, mainly in the choice of discrete methods in both temporal and spatial directions. Theoretical analysis can also be obtained on the convergence and stability of such methods presented in papers of the above sub-topics.
The aim of this special issue is to highlight the new developments in the area. The guest editors of this special issue would encourage our readers, while enjoying the remarkable results presented in this special issue, to participate in the many research activities in constructing highly accurate and structure-preserving methods, and to continue supporting and promoting the study of such strategies up to a new higher level in the near future.
Finally, the guest editors of this special issue would like to take this opportunity to thank many colleagues, including the IJCM editorial team, who have been helped, discussed, and supported the publication of this special issue. Last, but not least, the guest editors wish to thank their families for the marvellous understanding and support in fulfilling their research and editorial goals.