Question

Explain in detail Lax-wendroff technique. (CFD)

Answer 1

Lax-wendroff technique is mostly used for time and space marching solutions especially to hyperbolic equations. It is a explicit FDM method.

For example we will take a flow field problem governed by a hyperbolic equation.

Time marching solution of a an invicid flow using unsteady Euler equation.  

For demonstrating we will consider an easy Unsteady, 2D inviscid flow

Assuptions :

1. No body forces
2. No volumetric heat addition

Non conservative GDE is given as :

$Continuity : \frac{\partial\rho }{\partial t}= -\left ( \frac{\partial (\rho u)}{\partial x}+\frac{\partial(\rho V) }{\partial y} \right )== -\left ( \rho \frac{ \partial u}{\partial x}+u\frac{\partial\rho }{\partial x}+\rho \frac{\partial V}{\partial y}+V\frac{\partial \rho }{\partial y} \right )$$x.momentum: \frac{\partial u}{\partial t}=-\left ( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\frac{1}{\rho }\frac{\partial p}{\partial x} \right )$

ду y.momentum : — и ду ди + V— дх ду + 1др рду, at

$Energy : \frac{\partial e}{\partial t}=-\left ( u\frac{\partial e}{\partial x}+V\frac{\partial e}{\partial y}+\frac{p}{\rho }\frac{\partial u}{\partial x}+\frac{p}{\rho }\frac{\partial V}{\partial y} \right )$

We have to find a numerical soulution using time marching approach

Lax-wendroff technique is based on Taylor series expansion in time.

select any dependent flow variable from $\rho, u, V$

density at (i,j) at time $t+\Delta t$

ا-لرة Ду زران ذرا liti; DY isti Дх sx 2 t+st (et) Р + ... &'+) at com 2 زرا Hence it is assurs ed that flow field at time is known will give This at ation new frow Field equ time tt at similar write Now for uve expression

Grid pattern at time

t

At t t 2. t 2_ A+ ن) + (2) at ); 2. dt² 2 + 1 t t + (ot)? مد dt 2. dt² 2 ای () * () + ه عربی) + * () + ۔ ع (ع) + (2) ۔ على + +AE t t t 1 de At ره) + dt² 2. زا ttst Pis = P. At + ژرا (3) (ot)? + dt. درا 2دى 2

Continuity ap بول + u - ( de doc t e dy tv de doc dy at t t t t t t درا+نا Ui-I,; + t U. Pit,j Pity's - P. รไว้ = درا 2A 2 DX t t t t P. opo t + P.. Vi, '-1 i, Jol + Vijt! 2 DY Pijti 2 AY الا by differentiating continuity equation, Te + U. de + + d²p dt² = dre dxdt du de da dt dp du dx at Р doc.at de dy t لاله + par + ap dt + V. dy.at dy at dy.at dy

2 뢰 doc.at du at dx V + u du dt du doc du dy le dP doc 2 2 2 du C du du + vde J + doc? 은 dac.at d2 doc.dy dy doc as de p² dx dac + P 2L dx 신

t t t tu t 20: + درا -4 زرا- زرا+ دره 2 درا + 2²u 243 دراسل 2(دج) dacat t - زرا+ اد زرا ۔ - راسل + ا+را+نا + + A (As)@y) د- و اراده - درد t t u. t u. V د حد 2 لا 2 t t درا - 3 + زر 2 2 P. درن+ لما 2 2(د2) ہے ۔ t + + ہ 2. درا- تگ راس زرا- ا از راه 24x 2 A م)

Flow VARIABLE ttst CALCULATED list time ذرا۔ i,j it, j Flow VARIABLE Known | -ند t ^x

and at $t+\Delta t$ is given in the picture above picture.

So when you know the fow variable at t, you can find the values at $t+\Delta t$

Here actually we can directly calculate the flow variable along time direction. No need to solve any set of linear equation. But the method involved is length.

It will give a second order accurate in both space and time

In brief here idea is straight. But procedure is very lengthy.

Hey i thought of giving a typed soulution, but as its a very long one and i was going out of time thats why i switched to hand writing. I know its bit difficult to go through handwriting. Sorry for that. If you have any doubt feel free to comment. I will clarify...If you find this useful dont forget to give me a thumbs up ....Cheers buddy..

Have a nice day