1 Lecture 11: Unsteady Conduction Error Analysis

2 Last Time… l Looked at unsteady 2D conduction problems l Discretization schemes »Explicit »Implicit l Considered the properties of this scheme

3 This Time… l Complete unsteady scheme discussion »Properties of Crank-Nicholson scheme l Truncation error analysis

4 Unsteady Diffusion l Governing equation: l Integrate over control volume and time step

5 Discrete Equation Set l Dropping superscript (1) for compactness l Notice old values in discrete equation l To make sense of this equation, we will look at particular values of time interpolation factor f (f = 0, 1, 0.5)

6 Crank-Nicholson Scheme l Crank-Nicholson scheme uses f=0.5  t t1t1 t0t0 00 11 Linear profile assumption across time step

7 Crank-Nicholson: Discrete Equations l Linear equation set at current time level

8 Properties of Crank-Nicholson Scheme l Linear algebraic set at current time level – need linear solver l When steady state is reached,  P =  0 P. In this limit, steady discrete equations are recovered. »Steady state does not depend on history of time stepping »Will get the same answer in steady state by time marching as solving the original steady state equations directly l Will show later that truncation error is O (  t 2 )

9 Properties (cont’d) l What if l Crank-Nicholson scheme can be shown to be unconditionally stable. But if time step is too large, we can obtain oscillatory solutions

10 Truncation Error: Spatial Approximations l Face mean value of J e represented by face centroid value l Source term represented by centroid value: l Gradient at face represented by linear variation between cell centroid: l What is the truncation error in these approximations?

11 Mean Value Approximation l Consider 1-D approximation and uniform grid. l What is the error in representing the mean value over a face (or a volume) by its centroid value? l Expand in Taylor series about P

12 Mean Value Approximation (cont’d) l Integrate over control volume:

13 Mean Value Approximation (cont’d) l Complete integration to find: l Rearranging: l Thus is a second-order accurate representation

14 Gradient Approximation l To find expand about ‘e’ face l Subtract to find: l Linear profile assumption => second-order approximation

15 Temporal Truncation Error: Implicit Scheme l Cell centroid value (  ) = (  ) P at both time levels »Same as the mean value approximation »Second-order approximation l Value of  P 1 prevails over time step l Source term at (1) prevails over time step l What is the truncation error of these two approximations?

16 Truncation Error in Implicit Scheme (cont’d) l Consider a variable S(t) which we want to integrate over the time step: l Expand in Taylor series about new time level (1):

17 Truncation Error in Implicit Scheme (cont’d) l Integrate over time step: l Thus, representing the mean value over the time step by is a first-order accurate approximation

18 Closure In this lecture, we: l Completed the examination of unsteady schemes: »Crank-Nicholson l Looked at truncation error in various spatial and temporal approximations