Using pseudo-arclength continuation to trace the resonances of the Schrödinger equation

Size-dependent pull-in instability of electrostatically actuated microbeam-based MEMS

“…However, the notable challenge is that with the size effect now being included, the governing equation (7), unlike the classical one, is a sixth-order PDE and considerably more complicated. Both the GDQ method [31,32] and pseudoarclength algorithm [33,34] are required to discretize and to iterate the numerical solution (the combination of these two methods is, to our knowledge, for the first time applied in this note). The essence of the GDQ method is that the partial derivative of a function with respect to a variable is approximated by a weighted sum of function values at all discrete points in that direction.…”

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

“…Numerical bifurcation analysis of flow problems governed by the discretized Navier‐Stokes equations with a large number of degrees of freedom is still a challenging task. A review of the existing numerical bifurcation methods in fluid dynamics was proposed in the work of Dijkstra et al Toolbox exists for ordinary differential equations, such as MatCont and AUTO‐07p, hence with a small number of degrees of freedom . A large‐scale toolbox for systems of discretized partial differential equations is the library of continuation algorithms LOCA as part of the Trilinos framework…”

“…A review of the existing numerical bifurcation methods in fluid dynamics was proposed in the work of Dijkstra et al 1 Toolbox exists for ordinary differential equations, such as MatCont 2 and AUTO-07p, 3 hence with a small number of degrees of freedom. 4 A large-scale toolbox for systems of discretized partial differential equations is the library of continuation algorithms LOCA 5 as part of the Trilinos framework. 6 An alternative to the well-known incremental-iterative methods [7][8][9] is the asymptotic-numerical method (ANM).…”

Exploring ground states and excited states of spin-1 Bose–Einstein condensates by continuation methods

“…We first briefly introduce the framework of PACM and refer to [2,6,21,22,24] for details. Then we discuss how to determine initial points that can lead to ground state solutions of spin-1 BEC.…”

“…Substituting these dimensionless variables into (6) and then dividing by M a x 2 ffiffiffiffiffiffiffiffi Nd s p , the following dimensionless form of coupled GPE can be obtained ðl þ kÞw 1 ¼ e H n w 1 þ g s ðn 1 þ n 0 À n À1 Þw 1 þ g s w À1 w 2 0 ;…”