Numerical Modelling in Fortran: day 9. Paul Tackley, 2017

Transcription

1 Numerical Modelling in Fortran: day 9 Paul Tackley, 2017

2 Today s Goals 1. Implicit time stepping. 2. Useful libraries and other software 3. Fortran and numerical methods: Review and discussion. New features of Fortran 2003 & Projects: Define goals, identify information needed from me. Due date: 16 February 2018

3 Implicit time-stepping Diffusion equation again T t = 2 T

4 So far we have used explicit time stepping physical equation T t = 2 T explicit => calculate derivatives at old time T new T old Δt = 2 T old T new = T old + Δt 2 T old each Tnew can be calculated from already-known Told But: the value of del2(t) changes over dt

5 Implicit: calculate derivatives at new time T new T old Δt = 2 T new Put knowns on right-hand side, unknowns on LHS T new Δt 2 T new = T old More complicated to find Tnew: del2 term links all Tnew points. Why use it? Because the timestep is not limited: stable for any dt Eqn looks like Poisson s equation: we can modify existing solver

6 How about accuracy? Simple implicit and explicit methods are both first order accurate. Several related methods give second-order accuracy, including predictor-corrector, Runge- Kutta and semi-implicit. Of these, only the semi-implicit method has an unlimited time step, so let s focus on that! The semi-implicit method uses the average of derivatives at the beginning and end of the time step

7 semi-implicit diffusion T new T old Δt = 2 T new + 2 T old 2 Now generalised equation to deal with all 3 cases: explicit (beta=0), implicit (beta=1), semi-implicit (beta=0.5) T new T old Δt = β 2 T new + (1 β) 2 T old Put knowns on right-hand side, unknowns on LHS T new βδt 2 T new = T old + Δt(1 β) 2 T old

8 Rearrange to look like Poisson T new βδt 2 T new = T old + Δt(1 β) 2 T old ( 2 1 (βδt))t new = 1 βδt T old + Δt(1 β) 2 T old So, modify the Poisson solver to solve ( 2 C)T = f

9 Finite-difference modified Poisson ( 2 C)T = f T i 1, j + T i+1, j + T i, j 1 + T i, j+1 (4 + Ch 2 )T i, j h 2 = f i, j Iterative correction: R i, j = T i 1, j + T i+1, j + T i, j 1 + T i, j+1 (4 + Ch 2 )T i, j h 2 f i, j T n+1 = T n + αh 2 ( 4 + Ch 2 ) R

10 Optional 9 th homework (improves your grade if better than one of HW1-8) Modify your favourite program (either diffusion, advection-diffusion, infinite-pr convection or low- Pr convection) to include implicit diffusion Test for different values of beta (0, 0.5 and 1.0) If beta 0.5, ignore the diffusion timestep constraint! Due date: 8 January (2 working weeks from now)

11 Implementing implicit diffusion Modify your Poisson solver (iterations and multigrid) passing the constant C in as an extra argument. C=0 should give same result as before. Useful for streamfunction calculation. Add beta to the namelist inputs Modify diffusion calculation if beta>0 call multigrid, otherwise explicit like before use beta when calculating rhs term ignore diffusive timestep if beta 0.5 Run some tests with beta=0, 0.5 or 1.0 and see if you can tell the difference. It will have more effect at low Ra (or B), when diffusion dominates

12 Example: Low Pr convection Temperature equation with variable beta: T new βδt 2 T new = T old + Δt ((1 β) 2 T old ( v! T ) ) old Rearrange: ( 2 1 (βδt))t new = 1 βδt ( ) ( ) old T old + Δt (1 β) 2 T old v! T Call the new modified Poisson solver ( 2 C)u = f With (u=t) C = 1 (βδt); f = 1 βδt ( ) ( ) old T old + Δt (1 β) 2 T old v! T

13 Example: Low Pr convection (2) ω new Pr βδt 2 ω new = ω old + Δt Pr(1 β) 2 ω old v! ω Vorticity equation with variable beta: Rearrange: 2 1 Pr βδt ω new = 1 Pr βδt ω old + Δt Pr(1 β) 2 ω old v! ω ( ) old RaPr T old x ( ) old RaPr T old x Call the new modified Poisson solver ( 2 C)u = f With (u=w) C = 1 Pr βδt ; f = 1 Pr βδt ω old + Δt Pr(1 β) 2 ω old v! ω ( ) old RaPr T old x

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15 Note about boundary conditions If your existing iteration routines assume boundary conditions are 0 all round, they need modifying for T field. T boundary conditions are 0 at top, dt/dx=0 at sides, and On fine grid: 1 at bottom On coarse grids: 0 at bottom Either pass a boundary condition switch into the iteration routines, or write different routines for S and T fields.

16 Hand in Source code Results of test case(s) run using different values of beta and (if beta>0.5) different time steps including Time step > diffusive time step

17 Useful libraries and other software For standard operations such as matrix solution of systems of linear equations, taking fast- Fourier transforms, etc., it is convenient to download suitable routines from some free library, rather than writing your own from scratch. There are also more specialised free programs available for some applications A list is at: Much is available at:

18 Some well-known libraries lapack95: common linear algebra problems: equations, least squares, eigenvalues etc. also a parallel version scalapack MPI: for running programs on multiple CPUs. 2 common versions are MPICH: Open MPI: PETSc: Portable, Extensible Toolkit for Scientific Computation:

19 Project (1 KP)

20 Project (optional, 1 KP) 1. Chosen topic, agreed upon with me (suggestions given, also ask the advisor of your MSc or PhD project). Due end of Semesterprüfung (17 Feb 2017) Decide topic by final lecture! ct.html

21 Project: general guidelines Choose something either related to your research project and/or that you are interested in Effort: 1 KP => 30 hours. About 4 days work. I can supply information about needed equations and numerical methods that we have not covered

22 Some ideas for a project Involving solving partial differential equations on a grid (like the convection program) Wave propagation Porous flow (groundwater or partial melt) Variable-viscosity Stokes flow Shallow-water equations 3-D version of convection code Involving other techniques Spectral analysis and/or filtering Principle component analysis (multivariate data) Inversion of data for model parameters N-body gravitational interaction (orbits, formation of solar system,...) Interpolation of irregularly-sampled data onto a regular grid

23 History Fortran Review 60 years since first compiler versions 66, 77, 90, 95, 2003, 2008 Variables types: real, integer, logical, character, complex Implicit types and implicit none Initialisation parameters defined types precision (32 vs 64 bit etc.)

24 Fortran Review (2) Arrays fixed vs. automatic vs. allocatable vs. assumed shape index ranges data, reshape built-in array algebra Input/Ouput open, print, read, write, close formats iostat, rewind, status namelist binary vs. ascii, direct vs. sequential

25 Fortran Review (3) Flow control structures do loops (simple, counting, while), exit, cycle if elseif endif blocks select case forall where

26 Fortran Review (4) Functions and subroutines (procedures) Intrinsic functions: mathematical, conversion, internal (contains), external (& interface blocks), in modules array functions recursive functions and result statement named arguments optional arguments generic procedures overloading user-defined operators save

27 Fortran Review (5) Modules use public and private variables and functions => Pointers to variables, arrays, array sections, areas of memory, in defined types Optimization maximize use of cache and pipelining loops most important 90/10 rule (focus on bottlenecks) avoid branches, maximize data locality, unroll, tiling, Libraries, makefiles

28 For a more detailed review & discussion, see: 01/162125/fortran_class.pdf

29 New features of Fortran 2003 For a detailed description see: ftp://ftp.nag.co.uk/sc22wg5/n1601- N1650/N1648.pdf For a summary: ures.pdf Enhancements to derived types Parameterized derived types Improved control of accessibility Improved structure constructors Finalizers

30 New features of Fortran 2003 (2) Object-oriented programming support Type extension and inheritance Polymorphism Dynamic type allocation Type-bound procedures Data manipulation enhancements Allocatable components Deferred type parameters VOLATILE attribute Explicit type specification in array constructors & allocate statements Extended initialization expressions Enhanced intrinsic procedures

31 New features of Fortran 2003 (3) Input/output enhancements Asynchronous transfer Stream access User specified transfer ops for derived types User specified control of rounding Named constants for preconnected units FLUSH Regularization of keywords Access to error messages Procedure pointers

32 New features of Fortran 2003 (4) Support for the exceptions of the IEEE Floating Point Standard Interoperability with the C programming language Support for international characters (ISO byte characters...) Enhanced integration with host operating system Access to command line arguments,environment variables, processor error messages Plus numerous minor enhancements

33 Fortran 2008 Minor update of Fortran 2003 Brief summary (Wikipedia) Submodules additional structuring facilities for modules; supersedes ISO/IEC TR 19767:2005 Coarray Fortran a parallel execution model The DO CONCURRENT construct for loop iterations with no interdependencies The CONTIGUOUS attribute to specify storage layout restrictions The BLOCK construct can contain declarations of objects with construct scope Recursive allocatable components as an alternative to recursive pointers in derived types For more details see ftp://ftp.nag.co.uk/sc22wg5/n1801- n1850/n1828.pdf

34 Numerical methods review Approximations Concept of discretization Finite difference approximation Treatment of boundary conditions Equations diffusion equation Poisson s equation advection-diffusion equation Stokes equation and Navier-Stokes equation streamfunction and streamfunction-vorticity formulation Pr, Ra, Ek initial value vs. boundary value problems

35 Numerical methods Review (2) Timestepping stability (timestep, advection scheme) explicit vs. implicit, semi-implicit upwind advection Solvers direct iterative (relaxation), multigrid method Jacobi vs. Gauss-Seidel vs. red-black Parallelisation Domain decomposition Message-passing and MPI

36 Discussion We have focussed on one application (constant viscosity convection) Can apply same techniques to other applications / physical problems

37 Example: Darcy Equation (groundwater or partial melt flow) u = k η P gρy ˆ ( ) u=volume/area/time= Darcy velocity (really a flux) u = 0 k=permeability (related to porosity and interconnectivity) =viscosity of fluid, ρ =density of fluid, P=pressure in fluid η

38 Simplify and rearrange Take divergence: velocity is eliminated 2nd order equation for pressure Combine k and η into hydraulic resistance R u = k P + k η η gρ y ˆ = 0 k P = g η y 1 R ρk η P = g ρ y R A Poisson-like equation for pressure => easy to solve using existing methods

39 Example 2: Full elastic wave F=ma for a continuum: equation Most convenient to solve as coupled first-order PDEs Hooke s law for an isotropic material ε ij = 1 u i + u j 2 x j x i Where from equilibrium position µ λ,µ =shear modulus. constants Sometimes written Using K=bulk modulus ρ v t = σ σ ij = λδ ij ε kk + 2µε ij u=displacement of point are known as Lamé σ ij = Kε kk δ ij + 2µ(ε ij 1 3 ε kkδ ij )

40 Resulting equations to solve using finite differences Variables: 3 velocity components and 6 stress components On staggered grid (same as viscous flow modelling) t v x = 1 ρ x σ xx + y σ xy + z σ xz t σ xx = (λ + 2µ) v x x + λ v y y + v z z t v y = 1 ρ x σ xy + y σ yy + z σ zy t σ yy = (λ + 2µ) v y y + λ v x x + v z z t v z = 1 ρ x σ zx + y σ zy + z σ zz t σ zz = (λ + 2µ) v z z + λ v x x + v y y t σ xy = µ v x y + v y x t σ xz = µ v x z + v z x t σ yz = µ v y z + v z y

41 Mathematical classification of 2nd-order PDEs: Elliptical, hyperbolic, parabolic Elliptical 2 φ = f e.g., Poisson Parabolic φ t 2 φ e.g., diffusion Hyperbolic 2 φ 1 2 φ V 2 t 2 e.g., wave eqn. We ve done them all!

42 Waves: 2 P t 2 = v2 2 P Fluids: 1 v Pr t + v v = P + η v i + v j x j x i T t + v T = k T ( ) Percolation: k P = g η y ρk η + 1 Ek Ω v + Ra.T ˆ g v = 0 note the many similar terms! N-body: d 2 x i dt 2 = a i a i = G n,j i m j x j x. ( 2 j=1 i x j x i ) x j x i

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