Module 1: Introduction to Finite Difference …

1 Module 1: Introduction to Finite Difference Method and fundamentals of CFD Lecture 2: The Lecture deals with:Elementary Finite Difference Quotients Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 16/19/2012 4:36 PM Module 1: Introduction to Finite Difference Method and fundamentals of CFD Lecture 2: Elementary Finite Difference QuotientsFinite Difference representations of derivatives are derived from Taylor series example, if is the - component of the velocity, at point can beexpressed in terms of Taylor series expansion about point ( )Mathematically, Eq. ( ) is an exact expression for if the series practice, is small and any higher-order term of is smaller than . Hence, forany function Eq. ( ) can be truncated after a Finite number of :In terms of magnitude, and higher order are neglected, Eq. ( ) becomes( )Eq. ( ) is second-order accurate, because terms of order and higher have beenneglected.

2 If terms if order and higher are neglected, Eq. ( ) is reduced to( )Eq. ( ) is first-order Eqns. ( ) and ( ) the neglected higher-order terms represent the truncation , the truncation errors for Eqns. ( ) and ( ) are and It is now obvious that the truncation error can be reduced by retaining more terms in theTaylor series expansion of the corresponding derivative and reducing the magnitude of Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:16 PM. Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:16 PM Module 1: Introduction to Finite Difference Method and fundamentals of CFD Lecture 2: Elementary Finite Difference QuotientsLet us return to Eq. ( ) and solve for as: or( )In Eq. ( ) the symbol is a formal mathematical nomenclature which means terms oforder of , expressing the order the magnitude of the truncation error.

3 The first-order-accurate Difference representation for the derivative expressed by Eq. ( )can be identified as a first-order forward consider a Taylor series expansion for , and or( )Solving for , we obtain( )Eq. ( ) is a first-order backward expression for the derivative at grid point Subtracting Eq. ( ) from ( )( )And solving for from Eq. ( ) we obtain Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:16 PM( )Eq. ( ) is a second-order central Difference for the derivative at grid point Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:16 PM Module 1: Introduction to Finite Difference Method and fundamentals of CFD Lecture 2: Elementary Finite Difference QuotientsIn order to obtain a Finite Difference for the second-order partial derivative add Eq. ( ) and ). This produces( )Solving Eq.

4 ( ) for we obtain( )Eq. ( ) is a second-order central Difference form for the derivative at gridpoint Difference quotients for the derivatives are obtained in exactly the similar way. Theresults are analogous to the expression for the derivatives.[Forward Difference ][Backward Difference ][Central Difference ][Central Difference of secondderivative] Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 16/20/2012 12:17 PM Module 1: Introduction to Finite Difference Method and fundamentals of CFD Lecture 2: Elementary Finite Difference QuotientsCentral Difference given by Eq. ( ) can be interpreted as a forward Difference of thefirst order derivatives, with backward Difference in terms of dependent variables for thefirst-order derivatives. This is because or or The same approach can be made to generate a Finite Difference quotient for the mixedderivative at grid point.

5 Example,( )In Eq. ( ), if we write the derivative as a central Difference of derivatives,and further make use of central differences to find out the derivatives, we obtain ( ) Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:17 PMCongratulations, you have finished Lecture 2. To view the next lecture select it from the left hand sidemenu of the page or click the next button. Objectives_templatefile:///D:/chitra/npt el_phase2/mechanical/cfd/lecture2 of 26/20/2012 12:17 PM