Numerical Methods: Structured vs. unstructured grids. General Introduction: Why numerical methods? Numerical methods and their fields of application
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1 Numerical Methods: Structured vs. unstructured grids The goals o this course General : Why numerical methods? Numerical methods and their ields o alication Review o inite dierences Goals: Understanding the basics o all the inite s (dierences, elements, volumes The beauty: in the linear limit they are all really the same
2 Why numerical methods? Eamle: seismic wave roagation Seismometers homogeneous medium elosion In this case there are analytical solutions? Are they useul?
3 Why numerical methods? Eamle: seismic wave roagation Seismometers layered medium elosion... in this case quasi-analytical solutions eist, alicable or eamle or layered sediments...
4 Why numerical methods? Eamle: seismic wave roagation Seismometers long wavelength erturbations elosion in this case high-requency aroimations can be used (ray theory
5 Why numerical methods Eamle: seismic wave roagation Seismometers Generally heterogeneous medium elosion we need numerical solutions! we need grids! And big comuters
6 Partial Dierential Equations in Geohysics t P c s c ( s y z ressure acoustic wave seed sources The acoustic wave equation - seismology - acoustics - oceanograhy - meteorology t C k C v C RC C tracer concentration k diusivity v low velocity R reactivity sources Diusion, advection, Reaction - geodynamics - oceanograhy - meteorology - geochemistry - sedimentology - geohysical luid dynamics
7 Numerical methods: ields o alication Finite dierences - time-deendent PDEs - seismic wave roagation - geohysical luid dynamics - Mawell s equations - Ground enetrating radar -> robust, simle concet, easy to arallelize, regular grids, elicit method Finite elements - static and time-deendent PDEs - seismic wave roagation - geohysical luid dynamics - all roblems -> imlicit aroach, matri inversion, well ounded, irregular grids, more comle algorithms, engineering roblems Finite volumes - time-deendent PDEs - seismic wave roagation - mainly luid dynamics -> robust, simle concet, irregular grids, elicit method
8 Other Numerical methods: Particle-based methods - lattice gas methods - molecular dynamics - granular roblems - luid low - earthquake simulations -> very heterogeneous roblems, nonlinear roblems Boundary element methods - roblems with boundaries (ruture - based in analytical solutions - only discretization o lanes --> good or roblems with secial boundary conditions (ruture, cracks, etc Pseudosectral methods - orthogonal basis unctions - sectral accuracy o sace derivatives - wave roagation, GPR -> regular grids, elicit method, roblems with discontinuities
9 Numerical methods in all ields o Earth sciences Seismology Granular media - ruture Global dynamics Miing - Geochemistry Earthquake hysics Earth s magnetic ield Regional earthquakes
10 What is a inite dierence? Common deinitions o the derivative o (: lim 0 ( ( lim 0 ( ( lim 0 ( ( These are all correct deinitions in the limit ->0. But we want to remain FINITE
11 What is a inite dierence? The equivalent aroimations o the derivatives are: ( ( orward dierence ( ( backward dierence ( ( centered dierence
12 The big question: How good are the FD aroimations? This leads us to Taylor series...
13 Taylor Series... that leads to : 3 ( ( 1 ' '' ''' ( (! 3! (... ' ( O( The error o the irst derivative using the orward ormulation is o order. Is this the case or other ormulations o the derivative? Let s check!
14 Taylor Series... with the centered ormulation we get: 3 ( / ( / 1 ' ''' ( 3! (... ' ( O( The error o the irst derivative using the centered aroimation is o order. This is an imortant results: it DOES matter which ormulation we use. The centered scheme is more accurate!
15 Alternative Derivation o FD a a a ' d b b b ' d a b ( a b ( a b ' d Interolation Derivative a b 0 a b w1 0.5, w 0.5 Interolation weights w ' d 1, w 1 1 Derivative weights
16 Our irst FD algorithm! ( z y t s c P ressure c acoustic wave seed s sources Problem: Solve the 1D acoustic wave equation using the inite Dierence method. Solution: [ ] ( ( ( ( ( ( s t t c t
17 Problems: Stability ( t c [ ( ( ( ] ( t ( t s Stability: Careul analysis using harmonic unctions shows that a stable numerical calculation is subject to secial conditions (conditional stability. This holds or many numerical roblems. c ε 1
18 Problems: Disersion ( t c [ ( ( ( ] ( t ( t s True velocity Disersion: The numerical aroimation has artiicial disersion, in other words, the wave seed becomes requency deendent. You have to ind a requency bandwih where this eect is small. The solution is to use suicient grid oints er wavelength.
19 Our irst FD code! ( t c [ ( ( ( ] ( t ( t s % Time steing or i1:nt, end % FD dis(srint(' Time ste : %i',i; or j:n-1 d(j((j1-*(j(j-1/^; % sace derivative end new*-oldd*^; % time etraolation new(n/new(n/src(i*^; % add source term old; % time levels new; (10; % set boundaries ressure ree (n0; % Dislay lot(,,'b-' title(' FD ' drawnow
20 Our irst FD code! ( t c [ ( ( ( ] ( t ( t s % Time steing or i1:nt, % FD dis(srint(' Time ste : %i',i; Eercises (FD: 1. Increase the time ste manually and determine the stability limit numerically (c*/.. Make the medium heterogeneous. Put in a velocity contrast along the ais with a 30% erturbation. 3. Perturb the medium (e.g. 0% with random erturbations (dcrand([1 n]. What eect do you have on the roagating ulse? Do you think the result is accurate? 4. Comare the results with a simulation with time stes. or j:n-1 d(j((j1-*(j(j-1/^; % sace derivative end new*-oldd*^; % time etraolation new(n/new(n/src(i*^; % add source term old; % time levels new; (10; % set boundaries ressure ree (n0; end % Dislay lot(,,'b-' title(' FD ' drawnow
21 Finite Dierences - Summary Deending on the choice o the FD scheme (e.g. orward, backward, centered a numerical solution may be more or less accurate. Elicit inite dierence solutions to dierential equations are oten conditionally stable. The correct choice o the sace or time increment is crucial to enable accurate solutions. Sometimes it is useul to emloy so-called staggered grids where the ields are deined on searate grids which may imrove the overall accuracy o the scheme.
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