the expression
∆t <60∆W/a0.
∆W is the smallest grid size employed in the study and a0, is the speed of sound.”
It is unclear whether the CFL stability constraint is due to the boundary treatment or the Beam and Warming method itself.
The boundary condition difficulties that exist for many ADI schemes can usu- ally be traced back to a simple fact: the intermediate unknowns that arise from the splitting are not necessarily consistent approximations of the physical solution.
A notable exception in this regard is the splitting procedure developed by Douglas and Gunn [32]. Although not mentioned in the original papers, it was later under- stood [12] that the Douglas-Gunn splitting (with formal order of accuracy s = 2) yields equations for the intermediate unknowns that approximate the original PDE to order s−1 = 1. It follows that using the physical boundary conditions attn+1 for the intermediate steps preserves the order of accuracy of the method.
This thesis proposes ADI methods that address and overcome all the extant chal- lenges to ADI-based solvers. The underlying BDF multistep method together with BDF-like extrapolation for the nonlinear terms provides higher-order-accurate meth- ods with quasi-unconditional stability. An extension of the Douglas-Gunn splitting to our context guarantees the correct order of accuracy even in the presence of general boundary conditions.
domain decomposition is the natural basis for dividing the computational workload in a parallel implementation of a numerical solver. In this section we give a brief history of some domain decomposition strategies relevant to the one presented in this thesis.
1.4.1 The Schwarz alternating method
The earliest contribution to domain decompositions for partial differential equations is also the foundation of most modern domain decomposition solution strategies—
namely, the Schwarz method [75] which Schwarz developed for the same reason as given in point 1) in the introduction to this section—to solve a problem on a complex domain by using known solution methods on simpler ones. Here we give a brief history of the Schwarz method; see [40] for a more detailed account.
In his Ph.D. thesis, Riemann had taken for granted the existence of solutions to Laplace’s equation in general domains when he proved what would later be known as the Riemann mapping theorem. When it came to his attention, he invoked what is now called Dirichlet’s principle—that the solution of Laplace’s equation in a domainΩ with u=g on the boundary is given by the minimizer of the non-negative functional
J(u) = Z
Ω
1 2∇u2
among all twice-differentiable usatisfying the boundary conditions. However, Weier- strass showed with a counterexample that a non-negative functional need not attain a minimizer. Of course, the existence of harmonic functions was established for simple domains, like disks and rectangles. Schwarz used this fact to construct solutions in more complex geometries.
Ω1
Ω1∩Ω2 Ω2
Γ2
Γ1
∂Ω1
∂Ω2
Figure 1.2: A domain Ω given by the union of a diskΩ1 and a rectangle Ω2. For example, consider the problem
∆u= 0 inΩ u=g on∂Ω
where the domain Ω is given by the union of two sub-domains Ω1 and Ω2 such that Ω1 ∩Ω2 6= {∅}, as shown in Figure 1.2. (The disk and rectangle geometry is the example Schwarz himself used in his paper.) Let Γ1 =∂Ω1 ∩Ω2 and Γ2 =∂Ω2∩Ω1. The original Schwarz method produces the two sequences uk1 and uk2 given by the solutions of the sub-problems
∆uk+11 = 0 in Ω1
uk+11 =g on ∂Ω1∩∂Ω uk+11 =uk2 on Γ1
∆uk+12 = 0 inΩ2
uk+12 =g on∂Ω2∩∂Ω uk+12 =uk+11 onΓ2.
(1.8)
This is also known as thealternating Schwarz method. Notice that the solution of the second problem requires the solution of the first, so that the procedure is sequential.
Convergence of this method follows, in essence, from the maximum principle for harmonic functions.
In the early 1990’s, Lions formally introduced theparallel Schwarz method in [63],
which is a modification of the original method (1.8) given by the sub-problems
∆uk+11 = 0 in Ω1
uk+11 =g on ∂Ω1\Γ1
uk+11 =uk2 on Γ1
∆uk+12 = 0 in Ω2
uk+12 =g on ∂Ω2\Γ2
uk+12 =uk1 on Γ2.
(1.9)
Notice that the problems are independent, and thus form the basis for an elliptic PDE solver in a distributed computing environment.
1.4.2 Overset/Chimera/composite grid methods
More than 20 years before the work of Lions, Volkov made the first application of the Schwarz method to fully discrete PDEs in the method of composite meshes [92]—
indeed, Section 8 of that paper is entitled “The use of Schwarz’s alternating method for solving a system of difference equations.” This was also the first instance of a general class of methods which were developed around the same time and under different names—the composite mesh method [79], the Chimera grid method [80], and the overset grid method [13] being among the most common. In this thesis, we will use the latter of these terms.
The overset grid method is ideally suited for solvers relying on spatial discretiza- tions that make use of structured grids. Briefly, it involves decomposing the physical domain into a set overlapping logical rectangles, whereby a sub-problem is solved on each of the constituent grids. Data is communicated between these component grids by means of interpolation.
After the introduction of the overset grid method by Volkov and subsequent de- velopment by Starius [79], applications to CFD problems were explored by Steger, Dougherty, and Benek [80]. The method reached a state of maturity in the 1990s with the development of general purpose grid generation software, such as CMPGRD [21], later to evolve into the object oriented suite Overture [13], which also includes basic
capabilities for solving certain PDEs on overset grids. Of course, all the early work on overset methods was done in the context of finite differences.
More recently, the overset grid strategy has been successfully developed with the FC methodology for the solution of the compressible Navier-Stokes equations in two dimensions [2] and the elasticity equations in three dimensions [3]. A key development in those contributions is the extension of the FC method to overlapping “sub-patch”
block-decompositions of larger meshes. Although the contributions [2,3] have success- fully used the overset method in the context of explicit solvers, the goal of extending the framework to implicit and multi-domain implicit-explicit solvers has not yet been fully realized until now. This thesis presents the first steps toward a general frame- work for the solution of time-domain problems using multi-domain implicit-explicit FC solvers.