Theoretical advances. To illustrate our approach, consider the scalar ODE model,

Size: px

Start display at page:

Download "Theoretical advances. To illustrate our approach, consider the scalar ODE model,"

Anabel Easter Fields
5 years ago
Views:

1 Final Report GR/R69/ : Analysis of Numerical Methods for Incompressible Fluid Dynamics Personnel supported: Professor Philip Gresho (nominated Visiting Fellow; visited UK: 7/6/ 5//). Dr David Kay (visited UMIST: /7/ 6/7/; /9/ /9/; /6/ /6/ ) Introduction. Simulation of the motion of an incompressible fluid remains an important but very challenging problem. The resources required for accurate three-dimensional simulation of practical flows test even the most advanced of supercomputer hardware. The necessity for reliable and efficient solvers is widely recognised. The aim of this project was to study the performance of adaptive time-stepping methods and state-of-the-art iterative solvers in the context of numerical solution methods for the Navier-Stokes equations. In both research threads we have made important advances commensurate with those envisaged on the original proposal these are detailed below. The research effort was initiated and stimulated by Phil Gresho s six month visit to the UK. This visit was split roughly evenly between Dundee and Manchester, and discussions between Silvester, Griffiths and Gresho have continued electronically since Gresho s return to California. The main outcome of these technical exchanges has been a detailed analytic study of a particular adaptive time-stepping scheme that is a candidate best method in the context of low order finite element approximation in space. This work is currently being written up (in []), and a sample set of results is reviewed below. In parallel with this theoretical work, Silvester and Kay took the opportunity to discuss a variety of issues with Gresho associated with practical implementation of the adaptive time-stepping methodology. As a result, the approach has been implemented into a research code that is being developed by Kay at the University of Sussex. Whilst testing of this code is ongoing, the effectiveness of the methodology, in combination with our fast iterative solver technology, is already apparent. A discussion of practical issues is given in []. Theoretical advances. To illustrate our approach, consider the scalar ODE model,. y = λy y() = y ; (a) (b) with the simple solution y(t) = y e λt, () where λ is generally complex: λ = τ + i ω. (3) Here τ corresponds to a decay time constant and is meant to model simple diffusion, whereas the frequency, ω, is a simple model for advection. The combination of the two corresponds to advection-diffusion. The basic ODE method for our smart integration is the well known (second-order accurate, implicit) trapezoid rule (TR) chosen for several reasons: It is unconditionally stable (A-stable) stable when the ODE is. It is non-dissipative important when modelling pure advection (τ = ), or even advectiondominated problems ( τ ω). It is self starting.

2 There are no extraneous/spurious roots. Also relevant are the following facts: There are no explicit linear multi-step methods that are A-stable. The highest order A-stable method is two and the method is necessarily implicit. Of all second-order, A-stable methods, TR is the most accurate it has the smallest local truncation error coefficient. Applied to any ODE, the Trapezoid rule is simply y n+ = y n + t n ( ẏn + ẏ n+) where the ẏ n+ term on the right-hand-side makes the integrator implicit. Substituting the ODE for ẏ from (a) gives yn+ = λ t n + λ t n (4) y n. (5) At this point, if t n = t = constant, (5) merely represents the fixed- t version ( dumb version) of the integrator. Our real goal is a smart integrator that automatically changes t such that the local error is bounded for all time by a user-specified tolerance. To make progress towards this goal, we start from the truncation error term d n := y n+ y (t n+ ) = t3 n... y n +O ( t 4 ) n where y(t n+ ) is the exact solution at t = t n+, and match-up the Trapezoid rule with an appropriate explicit method that is the same order in terms of local truncation error. One possibility here is the second order Adams-Bashforth method, AB: ( y p n+ = y n + t n ẏ n + t n ẏn ẏ ) n, (7) t n where the superscript p signifies that it is a predictor it predicts a good estimate (for nonlinear equations) for y n+, the final (corrected) Trapezoid rule result. The basic idea here is that y p n+ leads to a technique that hopefully gives a good estimate of the otherwise unknown... y n+ in (6), from which local error control is easily developed. The first problem that we looked at is the heat equation (in R ): the aim is to compute u(x, t) satisfying the parabolic PDE u t = ν u x (8) on the unit interval, with zero initial data u(x, ), and with boundary data u(, t) =, u(, t) =. The motivation for studying (8) is that the PDE solution u can be analytically computed thus enabling us to compare numerical solutions with true physics. Taking a uniform linear finite element discretization in space (so that h = /N, with N = 64 (black) and N = 8 (red) representing the number of elements), the behaviour of the TR AB integrator with error tolerance tol = 5 is illustrated in Fig.. Notice the three distinct phases of the time-step evolution: The first phase is a rapid growth from the intial time-step t = 9, to a level fixed by the dominant eigenmode of the PDE problem. Specifically, t h exp(4t/3h )( tol) /3 (indicated with a solid line). (6)

3 Fig.. Heat equation ν = : log t vs. log t A second phase where the time-step follows the PDE dynamics; specifically t grows like t / independent of h. This is consistent with the notion of parabolic smoothing. A final phase (also independent of h) is where the time-step t (/π ) exp(π t/3)( tol) /3 (indicated in green), associated with the smallest eigenmode of the PDE problem. Note that if error control is tightened (by reducing tol) then the same qualitative behaviour is observed except that the timestep is reduced by a factor tol /3. Having established that the TR AB integrator is able to follow the physics of a diffusion problem, we moved on to the tougher case of advection-diffusion (again in R ): find u(x, t) satisfying u t = ν u x u x, (9) together with zero boundary data u(, t) = and u(, t) =. Although the exact form of the solution u is not available in this case, insight is nevertheless possible using classical asymptotic techniques (shown to us by a leading expert, Professor John Dold). A good example of what is possible is illustrated in Fig., where numerical and asymptotic solutions are compared in the case of a Gaussian wave profile being diffused and advected from left to right, prior to hitting a wall at the right hand boundary. We use a standard linear finite element discretization defined on a so-called Shishkin grid. (This is made up of two uniform grids of 64 elements, with a grid spacing chosen so that approximately half of the nodes lie in the boundary layer.) The agreement between numerics (dotted) and asymptotics (solid line) is impressive. The behaviour of our smart integrator with an error tolerance tol = 4 is illustrated in Fig. 3. There are four distinct phases of the time-step evolution in this case. Rapid growth from the intial time-step, t =, to a level fixed by the error tolerance and the discretization size h. A second phase where the time-step follows the (small-time) diffusion time-scale dynamics. As in Fig., t grows like t / independent of h (the straight line has slope /). A third phase (also independent of h) where the time-step is governed by the convection timescale dynamics. Here the profile is being convected from left to right, and the time-step stays essentially constant until the profile is wiped out at the right-hand boundary. 3

4 Fig.. Splat solution. ν = 4 : tol = 4 : t =.77. t = x BL solution x Fig. 3. Advection-diffusion equation ν = 4 : log t vs. log t. 3 4 t A final phase where the time-step rapidly grows (where the solution is essentially zero). Once again it is evident that the TR AB integrator is following the physics. An important issue that is considered in [] is the interplay between the temporal and the spatial discretization. Specifically, spectral properties of the discrete operators on stretched grids (favoured by Engineers) and Bakhvalov-Shishkin grids (favoured by mathematicians) have proved to be highly thought provoking both from an approximation viewpoint and a linear algebra perspective (the matrices are highly non-normal). We anticipate that the results in [] will be a fertile ground for both of these research communities in the future. Practical advances. Efficient implementation of the TR AB integrator into a code for solving the incompressible Navier-Stokes equation is the final aim of the project. Conventional codes typically use semi-implicit time integration leading to a Poisson or Stokes-type system at every time-step, but with a stability restriction on the time-step. In contrast, there is no such time-step restriction in our case. The price that must be paid for this improved robustness is the need to solve a so-called Oseen system (representing the linearised Navier-Stokes equations) at every time-step, see []. Initiated by David Kay s PDRA research (EPSRC grants GR/K96 and GR/L567), optimally efficient solvers for such Oseen problems have become a reality in the the last five years. (This is ongoing joint research involving Kay, Silvester and Howard Elman 4 t

5 from the University of Maryland, together with Daniel Loghin and Andy Wathen from Oxford University Computing Laboratory, see [3].) Specifically the preconditioning framework that we have developed offers the possibility of uniformly fast convergence independent of the problem parameters (namely; the mesh size, the time step and the Reynolds number). In contrast, conventional multigrid approaches to this problem tend to work well if the time-step is sufficiently small, but efficiency deteriorates rapidly if either the time-step or the Reynolds number is increased. Fig. 4. Snapshot of vortex shedding Re 4: tol = 3. Fig. 5. Effect of error tolerance on the flow solution. 3 Time step tol =.5 Time step tol = 3.5 Time step tol = 4 X Velocity Y Velocity Pressure

6 Guided by Phil Gresho s experience, we have tested Kay s code on a range of flow problems. One example is that of a flow being forced around a cylinder and ultimately reaching a periodic state of vortex shedding. (A snap-shot of the numerical solution in the shedding state is illustrated in Fig. 4.) Specific issues that we have addressed include alternative linearization approaches; the stability of the spatial discretization (adding streamline diffusion improves solver performance but can impinge on solution accuracy); mesh quality (big jumps in mesh-size need to be avoided in critical areas near the cylinder); and solver robustness with respect to the mesh-size and Reynolds number. Another point to note is that there is a delicate balance between the stopping criteria for the linear equation solver and the error control tolerance. The really fundamental issue is the need to balance spatial and temporal errors. This is illustrated by the results in Fig. 5, showing a point value of the velocity and pressure (on a fixed spatial mesh) with three choices of adaptive time-stepping tolerance. (Ideally we would like to automatically compute a tolerance so as to ensure that spatial error dominates.) If the error tolerance is not small enough ( 4 ) then the shedding frequency is wrong (cf. Fig. 4)! Dissemination aspects. Invited Talks given by David Silvester: Fast and robust solvers for time-discretised incompressible Navier-Stokes equations, Symposium on Finite Element Methods, Technical University Chemnitz, Germany, September ; Fast black-box preconditioners for self-adjoint PDE problems, Minisymposium on Numerical Linear Algebra, BAMC Meeting, University of Warwick, April, 5th Householder Symposium, Peebles, Scotland, June. Research Seminars given by David Silvester: Incompressible flow modelling is a dodgy business, University of Sussex, November ; A posteriori error estimation for elliptic PDEs, University of Manchester, February. Research Seminars given by Philip Gresho: Thermal convection in an 8: sidewall-heated enclosure, Scottish Computational Mathematics Symposium, Glasgow, September. On the stability of a particular steady incompressible flow at Reynolds number 8, University of Manchester, October ; A -D time periodic thermal convection problem, University of Manchester, October ; Some interesting aspects of a common mathematically ill-posed problem: impulsive starts of an incompressible fluid, UMIST, October ; How do to better CFD status and potential future prospects/directions for solving the incompressible Navier-Stokes equations, UMIST, November. (These seminars were widely advertised and well-attended. For the abstracts, see Publications: GR/R69/. Gresho, P., Griffiths, D. and Silvester, D. Adaptive time-stepping methods for solving incompressible CFD problems; part I: theoretical motivation, in preparation.. Kay, D. and Silvester, D. Adaptive time-stepping methods for solving incompressible CFD problems; part II: practical performance, in preparation. 3. Wathen, A., Loghin, D, Kay, D., Elman, H., and Silvester, D., A preconditioner for the 3D Oseen equations, Oxford University Computing Laboratory Report # /4,. 6

Fast solvers for steady incompressible flow

ICFD 25 p.1/21 Fast solvers for steady incompressible flow Andy Wathen Oxford University wathen@comlab.ox.ac.uk http://web.comlab.ox.ac.uk/~wathen/ Joint work with: Howard Elman (University of Maryland,