Ensemble Timestepping Algorithms for Natural Convection
This paper presents two algorithms for calculating an ensemble of solutions to laminar natural convection problems. The ensemble average is the most likely temperature distribution and its variance gives an estimate of prediction reliability. Solutions are calculated by solving two coupled linear systems, each involving a shared coefficient matrix, for multiple right-hand sides at each timestep. Storage requirements and computational costs to solve the system are thereby reduced. Stability and convergence of the method are proven under a timestep condition involving fluctuations. A series of numerical tests, including predictability horizons, are provided which confirm the theoretical analyses and illustrate uses of ensemble simulations.
Ensemble calculations are essential in predictions of the most likely outcome of systems with uncertain data, e.g., weather forecasting , ocean modeling , turbulence , etc. Ensemble simulations classically involve J sequential, fine mesh runs or J parallel, coarse mesh runs of a given code. This leads to a competition between ensemble size and mesh density. We develop linearly implicit timestepping methods with shared coefficient matrices to address this issue. For such methods, it is more efficient in both storage and solution time to solve J linear systems with a shared coefficient matrix than with J different matrices.
Prediction of thermal profiles is essential in many applications [1, 7, 17, 16]. Herein, we extend  from isothermal flows to temperature dependent natural convection. We consider two natural convection problems enclosed in mediums with: non-zero wall thickness  and zero wall thickness; Figure 1 illustrates a typical setup. The latter problem is often utilized as a thin wall approximation.
Consider the Thick wall problem. Let be polyhedral domains in with boundaries and , respectively, such that dist(,) . The boundary is partitioned such that with and . Given and for , let , , and satisfy
Here denotes the usual outward normal, denotes the unit vector in the direction of gravity, is the Prandtl number, is the Rayleigh number, and in and in is the thermal conductivity of the fluid or solid medium. Further, and are the body force and heat source, respectively.
Let and . To present the idea, suppress the spatial discretization for the moment. We apply an implicit-explicit time-discretization to the system (LABEL:s1) - (LABEL:s1f), while keeping the coefficient matrix independent of the ensemble members. This leads to the following timestepping method:
Consider the Thin wall problem. The main difference is a “” term on the r.h.s of the temperature equation (LABEL:s2T) absent in (LABEL:s1T). This apparently small difference in the model produces a significant difference in the stability of the approximate solution. In particular, a discrete Gronwall inequality is used which allows for the loss of long-time stability; see Section 4 below. Consider:
where is the first component of the velocity. If we again momentarily disregard the spatial discretization, our timestepping method can be written as:
By lagging both and the coupling terms in the method, the fluid and thermal problems uncouple and each sub-problem has a shared coefficient matrix for all ensemble members.
Remark: The formulation (LABEL:d1a) - (LABEL:d2a) arises, e.g., in the study of natural convection within a unit square or cubic enclosure with a pair of differentially heated vertical walls. In particular, the temperature distribution is decomposed into , where is the linear conduction profile and satisfies homogeneous boundary conditions on the corresponding pair of vertical walls.
In Section 2, we collect necessary mathematical tools. In Section 3, we present algorithms based on (LABEL:d1a) - (LABEL:d2a) and (LABEL:d1) - (LABEL:d2). Stability and error analyses follow in Section 4. We end with numerical experiments and conclusions in Sections 5 and 6. In particular, two stable, convergent ensemble algorithms are presented. These algorithms can be used to efficiently compute an ensemble of solutions to (LABEL:s1) - (LABEL:s1f) and (LABEL:s2) - (LABEL:s2f) and estimate predictability horizons. The ensemble average is shown to produce a better estimate of the energy in the system, for a test problem, than any member of the ensemble.
2 Mathematical Preliminaries
The inner product is and the induced norm is . Define the Hilbert spaces,
The explicitly skew-symmetric trilinear forms are denoted:
They enjoy the following continuity results and properties.
There are constants and such that for all u,v,w X and T,S , and satisfy
The proof of the first two identities is a calculation. The next four results follow from applications of Hölder and Sobolev embedding inequalities; see Lemma 2.2 on p. 2044 of . We will prove the last two results for ; for they are improvable. For all u,v,w X,
where Hölder, Ladyzhenskaya and Gagliardo-Nirenberg inequalities were used, respectively. Using the above result and inequalities and the first identity in Lemma LABEL:l1,
In similar fashion, there is a such that
The weak formulation of system (LABEL:s1) - (LABEL:s1f) is: Find , , for a.e. satisfying for :
Similarly, the weak formulation of system (LABEL:s2) - (LABEL:s2f) is: Find , , for a.e. satisfying for :
2.1 Finite Element Preliminaries
Consider a regular, quasi-uniform mesh of with maximum triangle diameter length . Further, for the system (LABEL:s1) - (LABEL:s1f), suppose that and lie along the meshlines of the triangulation of . Let , , and be conforming finite element spaces consisting of continuous piecewise polynomials of degrees j, l, and j, respectively. Moreover, assume they satisfy the following approximation properties :
for all , , and . Furthermore, we consider those spaces for which the discrete inf-sup condition is satisfied,
where is independent of . The space of discretely divergence free functions is defined by
The space , dual to , is endowed with the following dual norm
The discrete inf-sup condition implies that we may approximate functions in well by functions in ,
Suppose the discrete inf-sup condition (LABEL:infsup) holds, then for any
See Chapter 2, Theorem 1.1 on p. 59 of .
We will also assume that the mesh and finite element spaces satisfy the standard inverse inequality :
where denotes the minimum angle in the triangulation. A discrete Gronwall inequality will play a role in the upcoming analysis.
(Discrete Gronwall Lemma). Let , H, , , , and be finite nonnegative numbers for n 0 such that for N 1
then for all and N 1
See Lemma 5.1 on p. 369 of .
The discrete time analysis will utilize the following norms :
3 Numerical Scheme
Denote the fully discrete solutions by , , and at time levels , , and . Given , find satisfying, for every , the fully discrete approximation of the Thick wall problem:
Thin wall problem:
Remark: The treatment of the nonlinear terms in the time discretizations (LABEL:d1a) - (LABEL:d2a) and (LABEL:d1) - (LABEL:d2) leads to a shared coefficient matrix independent of the ensemble members.
4 Numerical Analysis of the Ensemble Algorithm
We present stability results for the aforementioned algorithms under the following timestep condition:
where . In Theorems LABEL:t1 and LABEL:t2, the nonlinear stability of the velocity, temperature, and pressure approximations are proven under condition (LABEL:c1) for the thick wall (LABEL:scheme:one:velocity) - (LABEL:scheme:one:temperature) and thin wall problems (LABEL:scheme:two:velocity) - (LABEL:scheme:two:temperature), respectively.
Remark: Stability of the numerical approximations can also be proven under: . If , then can be replaced with .
4.1 Stability Analysis
Consider the Thick wall problem (LABEL:scheme:one:velocity) - (LABEL:scheme:one:temperature). Suppose , . If (LABEL:scheme:one:velocity) - (LABEL:scheme:one:temperature) satisfy condition (LABEL:c1), then
Let in equation (LABEL:scheme:one:temperature) and use the polarization identity. Multiply by on both sides and rearrange. Then,
Use Cauchy-Schwarz-Young on ,
Consider . Add and subtract , use skew-symmetry, Lemma LABEL:l1, the inverse inequality, and the Cauchy-Schwarz-Young inequality. Then,
Using (LABEL:stability:thick:estg) and (LABEL:stability:thick:estbstar) in (LABEL:stability:thick) leads to
Let , add and subtract to the l.h.s. Regrouping terms leads to
By hypothesis, . Thus,
Sum from to and put all data on the right hand side. This yields
Therefore, the l.h.s. is bounded by data on the r.h.s. The temperature approximation is stable.
We follow an almost identical form of attack for the velocity as we did for the temperature. Let in (LABEL:scheme:one:velocity) and use the polarization identity. Multiply by on both sides and rearrange terms. Then,
Use the Cauchy-Schwarz-Young inequality on and and note that ,
Using skew-symmetry, Lemma LABEL:l1, the inverse inequality, and the Cauchy-Schwarz-Young inequality on leads to
Using (LABEL:stability:thick:estT), (LABEL:stability:thick:estf), and (LABEL:stability:thick:estb) in (LABEL:stability:thicku) leads to