Numerical Modelling in Fortran: day 10. Paul Tackley, 2016

Transcription

1 Numerical Modelling in Fortran: day 10 Paul Tackley, 2016

2 Today s Goals 1. Useful libraries and other software 2. Implicit time stepping 3. Projects: Agree on topic (by final lecture) (No lecture next week)

3 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:

4 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: ETH has a site license for the commercial NAG mathematical library: see ides.ethz.ch

5 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! FortranProject.html

6 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

7 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

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

9 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

10 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

11 How about accuracy? Simple implicit and explicit methods are both first order accurate. Several related methods give secondorder accuracy, including predictorcorrector, 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

12 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

13 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

14 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

15 9 th homework 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: 19 December (2 weeks from now)

16 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

17 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

18 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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20 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.

21 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