An unconditionally stable approximation of a circular flexible plate described by a fourth order partial differential equation

An unconditionally stable finite difference scheme for systems whose dynamics are described by a second-order partial differential equation is developed with use of regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization which was developed for first-order systems. The stability of the finite-difference scheme is analyzed by von Neumann's method. Using the new scheme, a discrete in time and space model of a deformable mirror is derived as the basis for control law design. The convergence of this scheme for various values of the discretization parameters is checked by numerical simulations.

[1]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[2]  Peter Steffen,et al.  Implicit discretization of linear partial differential equations and repetitive processes , 2009, 2009 International Workshop on Multidimensional (nD) Systems.

[3]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[4]  Anton Kummert,et al.  An approach to iterative learning control for spatio-temporal dynamics using nD discrete linear systems models , 2011, Multidimens. Syst. Signal Process..

[5]  Peter Steffen,et al.  Numerical iterative methods and repetitive processes , 2012, Multidimens. Syst. Signal Process..

[6]  Peter Steffen,et al.  Stability analysis for implicit second order finite difference schemes , 2011, The 2011 International Workshop on Multidimensional (nD) Systems.

[7]  Krzysztof Galkowski,et al.  Iterative learning control for spatio-temporal dynamics using Crank-Nicholson discretization , 2012, Multidimens. Syst. Signal Process..

[8]  S. BRODETSKY,et al.  Theory of Plates and Shells , 1941, Nature.

[9]  Andreas Rauh,et al.  An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems , 2016, Int. J. Appl. Math. Comput. Sci..

[10]  Krzysztof Galkowski,et al.  An unconditionally stable finite difference scheme systems described by second order partial differential equations , 2015, 2015 IEEE 9th International Workshop on Multidimensional (nD) Systems (nDS).

[11]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947 .