Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs

Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

“…where E is the energy density defined in (9). Notice that this local energy dissipation law is equivalent to (8).…”

“…In 2008, Wang et al [37] proposed the concept of local structure preservation for conservative PDEs. To date, the methodology of local structure preservation has been applied to solve a wide class of PDE systems [7,8,17,9,27].…”

“…However, it hardly works for the discrete case because we cannot derive the analogous local energy dissipation law with a discrete Leibnitz rule directly on the original PDE system. Hence, instead of rewriting the model into a first-order PDE system like in the previous works of Wang et al [37,7,8,17,9,27], we reformulate the model into an equivalent system with only second-order gradients by introducing intermediate variables. Based on the local energy dissipation structure of the equivalent system, we apply the implicit midpoint method in both space and time, to arrive at our first scheme.…”

Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

“…Proof. Multiplying equation (29) by D t A t p n− 1 j and equation (30) by D t A t q n− 1 j and then adding them together, we get…”

“…For the next few years, the theory of the local structure-preserving algorithm was used successfully for solving the PDEs (Refs. [23][24][25][26][27][28][29][30] and references therein), and the main advantage of the method is that it can keep the local structures of PDEs independent of boundary conditions.…”