Operator splitting implicit schemes for biharmonic problems arising in continuum mechanics
This work is devoted to application of operator splitting methods (known also as Alternating Direction Implicit, or ADI), to 2D problems involving fourth order spatial derivatives. This includes the equations with biharmonic operators which model the mechanics of elastic plates and stream-function formulation for viscous incompressible flows, e.g. complex thermoconvective flows. The original equations are rendered respectively into parabolic and ultraparabolic equations by adding time derivative with respect to an artificial time which allows implementation of ADI approach. Highly efficient, robust and accurate numerical schemes for evolutionary partial differential equations containing biharmonic spatial operators are devised. Our approach consists of complex treatments of the problems involving the following elements(steps): (i) devising a difference scheme; (ii) proving its stability and convergence; (iii) implementing the scheme in an algorithm: (iv) coding the algorithm, and (v) obtaining numerical results and analyzing their physical relevance.
In the first chapter, an ADI scheme for the Dirichlet problem for the parabolic equation containing fourth spatial derivatives is developed and used as an iterative procedure. A way to accelerate the convergence of iterations is proposed, allowing one to overcome the long-standing difficulties for bi-harmonic equations connected with the slow convergence of the iterative procedures based on the ADI method.
Chapter 2 treats the numerical investigation of the Navier-Stokes equations in terms of stream function whose formulation is not amenable to direct operator splitting. We use additional (fictitious) time alongside with the physical time rendering thus the model into an ultra-parabolic equation at each step with respect to the physical time. The convergence with respect to the fictitious time and absolute stability are proved. The scheme is implemented in a numerical algorithm in FORTRAN language. The effectiveness of the scheme is verified by numerical experiments for a model problem.
Chapter 3 treats a more complex physical phenomena: the unsteady natural convective flow (Boussinesq approximation) in a vertical slot with differentially heated walls and vertical temperature gradient. Nontrivial solutions are found for large Rayleigh numbers which show that with the increase of the stratification parameter, the mode of the instability changes from traveling-wave to stationary-wave.