Numerical Methods for PDEs

Similar documents
Partial Differential Equations

Math 257: Finite difference methods

Finite Dierence Schemes

Math 312 Lecture Notes One Dimensional Maps

Computational Fluid Dynamics. Lecture 3

The Advection-Diffusion equation!

Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations

Streamfunction-Vorticity Formulation

1. Linearization of a nonlinear system given in the form of a system of ordinary differential equations

1 6 = 1 6 = + Factorials and Euler s Gamma function

Numerical Methods for Partial Differential Equations

CONTENTS. Course Goals. Course Materials Lecture Notes:

Numerical Astrophysics: hydrodynamics

Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

Numerical Method for Blasius Equation on an infinite Interval

Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods

Løsningsførslag i 4M

Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw

Numerical Schemes to Solve Advective Contaminant Transport Problems with Linear Sorption and First Order Decay

3 Gauss map and continued fractions

An efficient time integration method for extra-large eddy simulations

Taylor polynomial solution of difference equation with constant coefficients via time scales calculus

PHY4905: Nearly-Free Electron Model (NFE)

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.

TMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods

Some properties of Boubaker polynomials and applications

Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method

Chapter 4. Fourier Series

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction

Solution of Linear Constant-Coefficient Difference Equations

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

6.3 Testing Series With Positive Terms

CHAPTER 1 SEQUENCES AND INFINITE SERIES

Numerical Methods for Partial Differential Equations

On Strongly Consistent Finite Dierence Approximations

A large class of initial value (time-evolution) PDEs in one space dimension can be cast into the form of a flux-conservative equation, u (19.1.

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

AE/ME 339 Computational Fluid Dynamics (CFD)

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.

A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS

Taylor expansion: Show that the TE of f(x)= sin(x) around. sin(x) = x - + 3! 5! L 7 & 8: MHD/ZAH

Pell and Lucas primes

Time-Domain Representations of LTI Systems

Chapter 10: Power Series

FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser

CS321. Numerical Analysis and Computing

Introduction to Signals and Systems, Part V: Lecture Summary

Fourier Series and the Wave Equation

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Differentiable Convex Functions

A Lattice Green Function Introduction. Abstract

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Chapter 4 : Laplace Transform

MATH 10550, EXAM 3 SOLUTIONS

SNAP Centre Workshop. Basic Algebraic Manipulation

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS

Solution of Differential Equation from the Transform Technique

Kinetics of Complex Reactions

DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS

INTEGRATION BY PARTS (TABLE METHOD)

Chapter 14: Chemical Equilibrium

Assignment Number 3 Solutions

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =

~j=zhax~ 6=0, t,2... ), k=~

MULTIPLE TIME SCALES SOLUTION OF AN EQUATION WITH QUADRATIC AND CUBIC NONLINEARITIES HAVING FRAC- TIONAL-ORDER DERIVATIVE

MAXIMALLY FLAT FIR FILTERS

Numerical Methods for Ordinary Differential Equations

Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)

Special Modeling Techniques

Mathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang

CS537. Numerical Analysis and Computing

IN many scientific and engineering applications, one often

Signals and Systems. Problem Set: From Continuous-Time to Discrete-Time

SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,

Some New Iterative Methods for Solving Nonlinear Equations

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

The Numerical Solution of Singular Fredholm Integral Equations of the Second Kind

Chapter 9: Numerical Differentiation

LOWER BOUNDS FOR THE BLOW-UP TIME OF NONLINEAR PARABOLIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!

Analysis of a Numerical Scheme An Example

1. Linear advection equations Let u R be a scalar unknown function of x R and t R + satisfying the Cauchy problem. u t + a(x, t)u x = 0, x R, t > 0,

AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS

Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco

Adaptation and Assessment of a High Resolution

MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of

Appendix: The Laplace Transform

Mathematical Series (You Should Know)

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia

A comparative study of a system of Lotka-Voltera type of PDEs through perturbation methods

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

Median and IQR The median is the value which divides the ordered data values in half.

Introduction to Optimization Techniques. How to Solve Equations

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

INTRODUCTORY MATHEMATICAL ANALYSIS

Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance

Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets

Transcription:

Numerical Methods for PDEs Hyperbolic PDEs: Coupled system/noliear coservatio laws/a oliear Lax-Wedroff scheme (Lecture 18, Week 6 Markus Schmuck Departmet of Mathematics ad Maxwell Istitute for Mathematical Scieces Heriot-Watt Uiversity, Ediburgh Ediburgh, February 19, 015 1/ 11

Outlie 1 Coupled system of equatios Noliear coservatio laws 3 Noliear coservatio laws: The Lax-Wedroff scheme / 11

Wave Equatio as Coupled Advectio Equatios We ca decouple the wave equatio u tt = a u xx as a pair of coupled first order equatios { u t + av x = 0 (CAE v t + au x = 0, which is supplemeted with the physical iitial coditios u(x, 0 ad v(x, 0. Check this: Differetiate the first equatio with respect to t, the secod with respect to x, the elimiate v xt Vector form of (CAE: u t + Au x = 0, where u = ( u v ( 0 a, A = a 0 3/ 11

Lax-Wedroff scheme (LW for (CAE Basic idea: Iclude dissipatio by a Taylor expasio i time up to d order ad use the relatio ( m ( t u = a m x u, such that u(x, t + k = u + ku t + 1 k u tt + O(k 3, (x,t = u Aku x + 1 A k u xx + O(k 3, (x,t u Ak D x h u + 1 A k δ x h u. This suggests the followig explicit scheme w +1 j = w j 1 P(w j+1 w j 1 + 1 P (w j+1 w j + w j 1 ( ( w u where w =, P = k z v h A. For I the uit matrix (LW w +1 j = (I P w j 1 P(I Pw j+1 + 1 P(I + Pw j 1. 4/ 11

Noliear coservatio laws So far: Liear hyperbolic problems May real problems: Require oliear equatios, e.g. fluid ad gas flow Noliear (AE: u t + a(uu x = 0 which icludes the Burgers equatio (BE u t + u u x = 0. Properties of (BE: Top moves faster tha the tail o either side ad evetually overtakes the tail formatio of a shock 5/ 11

u fastest faster steepes shock slow x flow directio This figure shows a flow i positive x-directio, i.e., a(u > 0. Coservatio law (CL: The umerical approximatio of u t + a(uu x = 0 leads ofte to ustable results. Rewritig it as follows (CL u t + a(uu x = 0 rearrage u t + (F (u = 0 x improves properties of associated umerical shemes where d F F(u = a(u, sice du x = df u du x. Example: Burgers equatio: a(u = u F (u = 1/u 6/ 11

Noliear (CL: A Lax-Wedroff scheme Goal: Exted the Lax-Wedroff method to oliear coservatio laws (NLCL u t + [F (u] x = 0. Idea: Apply the same strategy as for the advectio equatio: Step 1: Use a trucated Taylor series for u(x, t + k, i.e., u(x, t + k [u + k u t + 1 ] k u tt. (x,t Step : Replace the time derivatives by space derivatives, that is, u t = [F (u] x u tt = t ( [F (u] x = x ([F (u] t = ( F (u u t x = ( ( F (u[ F (u] x = F (u x x x F (u. 7/ 11

Step 3: Defiig leads to [ ] Q(u = F (u x F (u, so that u tt = x Q(u. u(x j, t +1 [ u k x F (u + 1 k ] x Q(u. (x j,t Step 4: Approximate x-derivatives by cetral differeces: x F (u F (w j+1 F (w j 1 = F j+1 F (xj,t h h x Q(u Q Q j+ 1 j 1, (xj,t h where we applied a spacig of h/ w.r.t. Q. j 1, 8/ 11

We have also used the otatio Q = F (w j+ 1 j+ 1 x F (u so x Q(u (xj,t (xj+ 1,t ( F F (w j+1 Fj j+ 1 h 1 { ( ( } h F (w F j+ 1 j+1 F j F (w F j 1 j Fj 1. Sice we do ot have values for u at half-iteger poits x j+ 1, we further approximate ad similarly for F (w. j 1 F (w 1 ( F (w j+ 1 j + F (wj+1 9/ 11

Step 5: Isertig these fidigs ito [ ] gives ( = wj Fj+1 F j 1 (NLW w +1 j a j 1 k h [ Fj Fj 1 ] } [ ] + {a k F h j+ 1 j+1 F j [ ] where a = 1 j+ 1 F (wj + F (wj+1 F (w. j+ 1 Remark: The (NLW scheme reduces to the liear (LW for F (u = au, a = cost, i.e. whe F = a. Check: w +1 j = wj k h [ ] } a awj awj 1 ( awj+1 aw j 1 + k h { [ ] a awj+1 aw j = wj p ( wj+1 w j 1 + p ( wj+1 w j + wj 1, p = ak h which is the formula derived earlier. 10/ 11

Stability of (NLW I the liear scheme we have stability oly for p 1. I the oliear scheme, F (u plays the role of a, so we should require that F (w j k h 1 for all m. For safety we replace 1 by = 0.9 ad require k = 0.9h max j F (w j Note that k will vary from step to step. 11/ 11

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback