Upwinding. Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

Transkript

1 p. 1/20 Upwndng home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/cover_sheet.tex page 1 of 20

2 p. 2/20 Table of contents 1. Introducton Upwndng (Transport Equaton) 2. Comparson wth the Central Dfference Scheme 3. Upwndng (Advecton Equaton) 4. Courant number 5. Alternatve Interpretaton 6. Further dscretzaton schemes home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/toc_9.tex page 2 of 20

3 p. 3/20 Introducton to upwnd methods Before we have manly consderd centered methods, symmetrc about the pont where we are updatng the soluton. These methods are approprate for all systems, n whch the nformaton s travellng unformly n all drectons. Dfferental equatons that are descrbng these knds of problems are called ellptc equatons. The model example s a stone that s thrown nto a lake. The waves are propagatng n crcles. The Laplace equaton of groundwater flow and the heat dffuson equaton are examples for ths knd of physccal problems. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/ntro_upwnd_1.tex page 3 of 20

4 p. 4/20 Introducton to upwnd methods For hyperbolc problems, however, we expect nformaton to propagate n a certan drecton. Supercrtcal flow n rvers or supersonc speed of an arplane are physcal examples. In an advecton/dffuson problem, e.g. represented by the transport equaton v c x D 2 c x = 0 2 we speak of a parabolc equaton. Ths s the type of equaton that s lyng between the two extremes (ellptc and hyperbolc). The advectve term n the equaton represents the hyperbolc character, whereas the dffuson term represents the ellptc part. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/ntro_upwnd_1b.tex page 4 of 20

5 p. 5/20 Introducton to upwnd methods The Peclet number Pe = vl D descrbes the relaton between the advectve and the dffusve part. If the Peclet number s very small the dffusve part domnates and central dscretzaton schemes are the best choce. If, on the ohter hand, the Peclet number s very hgh, then the advectve term s domnatng and Upwndng methods are approprate. Upwndng schemes use backward dfferences to dscretze the advectve term. The nformaton s travellng wth the flow and t therefore makes sense to take only the upstream nformaton nto account. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/ntro_upwnd_1c.tex page 5 of 20

6 p. 6/20 Comparson of upwndng and central scheme In the Upwndng method we consder the upstream and the current node n the advecton term: v c c 1 D c +1 2c + c 1 () 2 = 0 Central dfference scheme: v c +1 c 1 D c +1 2c + c 1 () 2 = 0 Take the upwndng scheme and add and subtract the advecton term of the central scheme. v c c 1 + v c +1 c 1 v c +1 c 1 D c +1 2c + c 1 () 2 = 0 home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/comp_upwnd_central.tex page 6 of 20

7 Comparson of upwndng and central scheme p. 7/20 v c c 1 v c +1 c 1 v 2c 2c 1 c +1 + c 1 + v c +1 c 1 D c +1 2c + c 1 () }{{ 2 } central scheme + v c +1 c 1 = 0 D c +1 2c + c 1 () 2 = 0 v c c c 1 v c +1 2c + c 1 2 () 2 v D c +1 2c + c 1 D 2 () 2 + v c +1 c 1 + v c +1 c 1 + v c +1 c 1 D c +1 2c + c 1 () 2 = 0 D c +1 2c + c 1 () 2 = 0 D c +1 2c + c 1 () 2 = 0 home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/comp_upwn_central2.tex page 7 of 20

8 Comparson of upwndng and central scheme p. 8/20 Pe 2 Dc +1 2c + c 1 () 2 + v c +1 c 1 D c +1 2c + c 1 = 0 () 2 ( D 1 + Pe ) c +1 2c + c 1 = 0 2 }{{} 2 The dffuson s artfcally ncreased. v c +1 c 1 v c x D(1 + Pe 2 ) 2 c x 2 = 0 Ths dffuson s referred to as numercal dffuson. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/comp_upwn_central3.tex page 8 of 20

9 p. 9/20 Combnaton of the methods One can also try to combne the advantages of the two methods. Use the better accuracy of the central dfference scheme wth the stablty of the upwnd scheme. The upwnd coeffcent α s the weghtng factor. Central dfferences c x centr = (u +1 u 1 ) Upwndng: c xupw = (u u 1 ) Lnear nterpolaton wth upwnd-coeffcent α c xadv = α (u u 1 ) + (1 α) (u +1 u 1 ) home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/comb_methods.tex page 9 of 20

10 p. 10/20 Advecton equaton In a scalar advecton problem, e.g. the hyperbolc advecton equaton c t + v c x = 0 (1) the velocty s ndependent of the concentraton. The lnear, hyperbolc frst-order equaton has been studed ntensvely and several schemes have been proposed to solve ths equaton. Here we wll present some of these. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/adv_eqn.tex page 10 of 20

11 p. 11/20 Upwndng (Advecton equaton) Usng an explct tme dscretzaton and a backward dfference scheme n space leads to the frst-order upwnd method for the advecton equaton 1 t ( u n+1 u n ) + v ( u n u n 1 ) = 0 t n+1 u n+1 t n u n 1 u n u n +1 x where the chaned lnes ndcate tme and space levels and the sold boxes are the control volumes. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/ntro_upwnd_3.tex page 11 of 20

12 p. 12/20 Upwndng II (Advecton equaton) The frst-order upwnd method for the advecton equaton s condtonally stable. The rato u n+1 = u n v t (un u n 1). Cr = v t s known as the Courant Number. It can be nterpreted as the physcal velocty v over the grd velocty / t. For Cr = 1 the soluton s exact, for smaller Courant numbers the soluton s stable but a large amount of numercal dffuson s ntroduced. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/ntro_upwnd_3.tex page 12 of 20

13 p. 13/20 Alternatve nterpretaton Thnk of u n as beng the value at the grd pont, u n ũ(x, t n ) n a fnte dfference settng. Then snce u(x,t) s constant along ts characterstc,.e. the path lne, we expect: u n+1 ũ(x,t n+1 ) = ũ(x v t,t n ) u n+1 t n+1 t n u n 1 u n x v t x t home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/altern_nterpr_1.tex page 13 of 20

14 p. 14/20 Alternatve nterpretaton If u n+1 represents the value at a grd pont, then we can trace the characterstc back and nterpolate. We approxmate ths value by a lnear nterpolaton between the grd values u n 1 and u n. u n+1 = v t un 1 + Ths s smply the upwnd method. ( 1 v t ) u n Here we can see that we have a weghted average between the two nodes. Ths makes physcal sense as long as the characterstc really ntersects the old tme level between the two nodes, whch s agan our condton Cr 1. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/altern_nterpr_2.tex page 14 of 20

15 p. 15/20 Central dfference scheme The unstable centered method appled to the advecton equaton wth an explct tme dscretzaton scheme yelds the followng equaton 1 t ( u n+1 u n+1 ) u n v ( ) + u n +1 u n 1 = 0. = u n v t ( ) u n +1 u n 1. Note the slght dfference n the advecton term. The addtonal factor 1 2 and the un +1 nstead of u n. Ths soluton s always unstable! home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/central_dff.tex page 15 of 20

16 p. 16/20 Lax Scheme I t unknown n+1 known t n 1 +1 s The dffusve (lax) method dstngushes tself from the unstable model nsofar as that for the tme dervaton not u n s used, but a mean from u n+1 1 and un Ths scheme s stable for Cr 1. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/lax_scheme_1.tex page 16 of 20

17 p. 17/20 Lax Scheme II u x un +1 u n 1 (as n the unstable method) u t 2un+1 u n 1 un +1 2 t (new compared to the unstable method) or, more general: u t un+1 [ αu n + (1 α) ( ) ] u n 1 + u n +1 /2 t (0 α 1; for α = 1 results the unstable model). u u n or u un 1 + u n +1 2 home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/lax_scheme_2.tex page 17 of 20

18 p. 18/20 Pressmann Scheme t k+1 Θ t ( 1 Θ ) t unknown known approxmated k x j 2 x j 2 j s j The Pressmann scheme dffers from the before mentoned schemes by also averagng the tme dervatve. The frst dervatves are depctable n the pont + n the space and tme drecton. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/presmann_1.tex page 18 of 20

19 p. 19/20 Pressmann Scheme II u + x u + t = u+ Θuk+1 x j u k+1 = u+ t = 1 2 t 1 t ( u k+1 j j ( u k+1 j + u k+1 2 u k j + uk+1 u k ). + (1 Θ) uk j u k, j ) uk j + u k 2 For the quantty of state n the calculaton pont, the mean value s establshed: u + Θ uk+1 j + u k+1 + (1 Θ) uk j + u k. 2 2 The method s stable for Θ > 0, 5. yelds best results between: Θ 0, 55 0, 6. and creates numerc dampng for larger values. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/presmann_2.tex page 19 of 20

20 p. 20/20 Further schemes Leap-frog u n+1 u n 1 2 t + v un +1 un 1 = 0. Lax Wendroff u n+1 u n t Beam-Warmng + v un +1 un 1 v 2 t 2 u n +1 2un +un 1 2 = 0. u n+1 u n t + v 3un 4un 1 +un 2 v 2 t 2 u n 2un 1 +un 2 2 = 0. Crank-Ncholson u n+1 u n t + θv un+1 +1 un (1 θ)v un +1 un 1 = 0. home/lehre/vl-mhs-1-e/folien/vorlesung/9_upwinding/further_schemes.tex page 20 of 20