2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13

Transcription

1 2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13 REVIEW Lecture 12: Classification of Partial Differential Equations (PDEs) and eamples with finite difference discretizations Parabolic PDEs Elliptic PDEs Hyperbolic PDEs Error Types and Discretization Properties: L ( ) 0, Lˆ ( ) ˆ 0 Consistency: L ( ) L ˆ ( ) 0 when 0 Truncation error: L ( ) Lˆ ( ) O( Error equation: L ( ) Lˆ ( ˆ ) L ˆ ( ) ˆ 1 Stability: Const. (for linear systems) L Convergence: L O( p ) ˆ 1 p ) for 0 (for linear systems) 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 1 1

2 2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13 REVIEW Lecture 12, Cont d: Classification of PDEs and eamples Error Types and Discretization Properties Finite Differences based on Taylor Series Epansions Higher Order Accuracy Differences, with Eamples Incorporate more higher-order terms of the Taylor series epansion than strictly needed and epress them as finite differences themselves (making them function of neighboring function values) If these finite-differences are of sufficient accuracy, this pushes the remainder to higher order terms => increased order of accuracy of the FD method General approimation: m s u au m i ji j ir Taylor Tables or Method of Undetermined Coefficients (Polynomial Fitting) Simply a more systematic way to solve for coefficients a i 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 2 2

3 FINITE DIFFERENCES Outline for Today Classification of Partial Differential Equations (PDEs) and eamples with finite difference discretizations (Elliptic, Parabolic and Hyperbolic PDEs) Error Types and Discretization Properties Consistency, Truncation error, Error equation, Stability, Convergence Finite Differences based on Taylor Series Epansions Higher Order Accuracy Differences, with Eample Taylor Tables or Method of Undetermined Coefficients (Polynomial Fitting) Polynomial approimations Newton s formulas Lagrange polynomial and un-equally spaced differences Hermite Polynomials and Compact/Pade s Difference schemes Boundary conditions Un-Equally spaced differences Error Estimation: order of convergence, discretization error, Richardson s etrapolation, and iterative improvements using Roomberg s algorithm 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 3 3

4 References and Reading Assignments Part 8 (PT 8.1-2), Chapter 23 on Numerical Differentiation and Chapter 18 on Interpolation of Chapra and Canale, Numerical Methods for Engineers, 2010/2006. Chapter 3 on Finite Difference Methods of J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3 rd edition, 2002 Chapter 3 on Finite Difference Approimations of H. Loma, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, Numerical Fluid Mechanics PFJL Lecture 13, 4 4

5 FINITE DIFFERENCES: Interpolation Formulas for Higher Order Accuracy 2 nd approach: Generalize Taylor series using interpolation formulas Fit the unknown function solution of the (P)DE to an interpolation curve and differentiate the resulting curve. For eample: Fit a parabola to data at points i1, i, i1 ( i i i1 ) differentiate to obtain: f( ) f( ) f( f '( i ) i1 i ( i i1 ), then This is a 2 nd order approimation i1 i i1 i1 i i1 i ) For uniform spacing, reduces to centered difference seen before In general, approimation of first derivative has truncation error of the order of the polynomial All types of polynomials or numerical differentiation methods can be used to derive such interpolations formulas Polynomial fitting, Method of undetermined coefficients, Newton s interpolating polynomials, Lagrangian and Hermite Polynomials, etc 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 5 5

6 FINITE DIFFERENCES Higher Order Accuracy: Taylor Tables or Method of Undetermined Coefficients Taylor Tables: Convenient way of forming linear combinations of Taylor Series on a term-by-term basis The Taylor table for a centered three point Lagrangian approimation to a second derivative. What we are looking for Taylor series at: j-1 j j+1 Sum each column starting from left, force the sums to zero and so choose a, b, c, etc 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 6 6

7 FINITE DIFFERENCES Higher Order Accuracy: Taylor Tables Cont d The Taylor table for a centered three point Lagrangian approimation to a second derivative. Sum each column starting from left and force the sums to be zero by proper choice of a, b, c, etc: a c b 0 a b c = Familiar 3-point central difference Truncation error is first column in the table that does not vanish, here fifth column of table: a c 4 u u j 12 j 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 7 7

8 FINITE DIFFERENCES Higher Order Accuracy: Taylor Tables Cont d The Taylor table for a backward three point Lagrangian approimation to a second derivative a2 a 1 3 u u a 2 a 1 b /2 and j 3 j (as in lecture 12) 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 8 8

9 Finite Differences using Polynomial approimations Numerical Interpolation: Newton s Iteration Formula Standard triangular family of polynomials Newton s Computational Scheme + Divided Differences By recurrence: 0 0 Second divided First differences divided differences 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 9 9

10 Finite Differences using Polynomial approimations Equidistant Newton s Interpolation Equidistant Sampling Divided Differences Equidistant Step size Implied Triangular Family of Polynomials Equidistant Sampling 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 10 10

11 Numerical Differentiation using Newton s algorithm for equidistant sampling Triangular Family of Polynomials Equidistant Sampling f() n=1 First order h 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 11 11

12 Numerical Differentiation using Newton s algorithm for equidistant sampling, Cont d Second order f() n=2 Forward Difference Central Difference h h Second Derivatives n=2 Forward Difference n=3 Central Difference 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 12 12

13 Finite Differences using Polynomial approimations Numerical Interpolation: Lagrange Polynomials (Reformulation of Newton s polynomial) f() 1 k-3 k-2 k-1 k k+1 k+2 Difficult to program Difficult to estimate errors Divisions are epensive Important for numerical integration 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 13 13

14 Hermite Interpolation Polynomials and Compact / Pade Difference Schemes Use the values of the function and its derivative(s) at given points k For eample, for values of the function and of its first derivatives at pts k n m u u ( ) a k ( ) u k b k ( ) k 1 k 1 k General form for implicit/eplicit schemes (here focusing on space) s m q u b i a u m i ji ir j i i p Generalizes the Lagrangian approach by using Hermitian interpolation Leads to the Compact difference schemes or Pade schemes Are implemented by the use of efficient banded solvers 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 14 14

15 FINITE DIFFERENCES: Higher Order Accuracy Taylor Tables for Pade schemes Taylor table for a central three point Hermitian approimation to a first derivative. u u u 1 d e ( auj 1 buj cu j 1)? j 1 j j 1 j j 1 2 u j j 1 j 1 j j+1 Image by MIT OpenCourseWare Numerical Fluid Mechanics PFJL Lecture 13, 15 15

16 FINITE DIFFERENCES: Higher Order Accuracy Taylor Tables for Pade schemes, Cont d Taylor table for a central three point Hermitian approimation to a first derivative. u u u 1 d e ( auj 1 buj cu j 1)? j 1 j j 1 Image by MIT OpenCourseWare. Sum each column starting from left and force the sums to be zero by proper choice of a, b, c, etc: a b c 0 a b c d e d e 0 Truncation error is first column in the table that does not vanish, here sith column: 4 5 u j 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 16

17 i( Compact / Pade Difference Schemes: Eamples We can derive family of compact centered approimations for up to 6 th order using: α ( φ ( i+1 + ( φ ( φ + α = β ( i-1 φ φ i+1 i γ φ φ i+2 i-2 4 Scheme Truncation error α β γ CDS-2 ( ( 2 3 φ 3! Comments: CDS-4 ' Pade-4 13 (. ( 4 5 φ 3 3! 5 ( ( 4 5 φ 5! Pade schemes use fewer computational nodes and thus are more compact than CDS Pade-6 ' 4 ( ( 6 7 φ 7! Can be advantageous (more banded systems!) Image by MIT OpenCourseWare Numerical Fluid Mechanics PFJL Lecture 13, 17

18 Higher-Order Finite Difference Schemes Considerations Retaining more terms in Taylor Series or in polynomial approimations allows to obtain FD schemes of increased order of accuracy However, higher-order approimations involve more nodes, hence more comple system of equations to solve and more comple treatment of boundary condition schemes Results shown for one variable valid for mied derivatives To approimate other terms that are not differentiated: reaction terms, etc Values at the center node is normally all that is needed However, for strongly nonlinear terms, care is needed (see later) Boundary conditions must be discretized 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 18 18

19 Finite Difference Schemes: Implementation of Boundary conditions For unique solutions, information is needed at boundaries Generally, one is given either: i) the variable: u(, t ) u ( t ) (Dirichlet BCs) bnd bnd ii) a gradient in a specific direction, e.g.: iii) a linear c ombination of the two quantities u ( bnd, t ) = bnd (t) (Neumann BCs) (Robin BCs) Straightforward cases: If value is known, nothing special needed (one doesn t solve for the BC) If derivatives are specified, for first-order schemes, this is also straightforward to treat 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 19 19

20 Finite Difference Schemes: Implementation of Boundary conditions, Cont d Harder cases: when higher-order approimations are used At and near the boundary: nodes outside of domain would be needed Remedy: use different approimations at and near the boundary Either, approimations of lower order are used Or, approimations go deeper in the interior and are one-sided. For eample, 1 st order forward-difference: u u 2 u u 2 u 1 ( bnd, t) 2 1 Parabolic fit to the bnd point and two inner points: ( ) u ( ) u ( ) ( ) u u u 3 4 u 2 3u 1 for equidistant nodes, ) ( )( )( 2 ) 2 ( bnd t Cubic fit to 4 nodes (3 rd order difference): u ( bnd, t ) 2u 4 9u 3 18u 2 11u1 3 O( ) for equidistant nodes 6 18u 2 9u 3 2u 4 6 u Compact schemes, cubic fit to 4 pts: u ( bnd, t ) u 1 for equidistant nodes In Open-boundary systems, boundary problem is not well posed => Separate treatment for inflow/outflow points, multi-scale approach and/or generalized inverse problem (using data in the interior) 2.29 Numerical Fluid Mechanics PFJL Lecture 13, 20 20

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