Full HTML for

Basic foilset Introduction to Physics and Numerics of NAS Benchmarks

Given by Geoffrey C. Fox at CPSP713 Case studies in Computational Science on Spring Semester 1996. Foils prepared 15 March 1996
Outside Index Summary of Material

This is first of three foilsets on CFD and NAS Benchmarks
This describes the four basic NAS benchmarks and their relation to the Navier Stokes Equations in the the 5 component CFD equations
We use opportunity to discuss time discretization and stepping in general based on Hirsch CFD book and Numerical Recipes
Stability, Implicit and Explicit formulations are introduced
Beam Warming Equations and their stability

Table of Contents for full HTML of Introduction to Physics and Numerics of NAS Benchmarks

 Denote Foils where Image Critical
Denote Foils where HTML is sufficient
1 CPS713 Case Study II) CFD and Numerical Relativity
NAS Part I -- Initial NAS Benchmark Setup
2 Abstract of CPS713-Case Study II
Initial NAS Benchmark Setup
3 NAS Parallel Benchmarks
4 Summary of Sec 3: NAS Parallel Benchmarks:The CFD Kernels
5 Overview of Four Computational (PDE Solver) Kernels in
NAS Parallel Benchmarks
6 First Two Computational Kernels
7 Second Two Computational Kernels
8 The First Two Simplifications of the NAS Parallel Benchmarks
9 The Second Two Simplifications of the NAS Parallel Benchmarks
10 Basic Formalism for 5 Component CFD
11 What is relation of NAS benchmarks to Navier Stokes Equations -- Terms E F G?
12 What is relation of NAS benchmarks to Navier Stokes Equations -- Terms T V W ?
13 What is relation of NAS benchmarks to Navier Stokes Equations-- Term H and Boundary Conditions ?
14 What is relation of NAS benchmarks to Navier Stokes Equations -- Space and Time Variables
15 Treatment of Time Discretization for PDE's
16 Treatment of Time Stepping for PDE's :
Numerical Recipes -- Chapter 19
17 Comparison of Different Methods for Solving Linear Convection Equation
18 Overview of Stability and Accuracy of Partial Differential Equation Solvers
19 Solution of Homogenous Equations by Fourier Analysis
20 Solution of Linear Stability Equations for the Linear Convection Equation
21 General Stability Analysis of some two-step Methods -- Basic Iteration Equation
22 Differential Operator and its Eigenvalues for Beam Warming Equations
23 Stability Conditions for Beam Warming Equations
24 The effect of Nonlinearity on Implicit Equations

Outside Index Summary of Material

HTML version of Basic Foils prepared 15 March 1996

Foil 1 CPS713 Case Study II) CFD and Numerical Relativity
NAS Part I -- Initial NAS Benchmark Setup

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Geoffrey Fox
NPAC
111 College Place
Syracuse NY 13244-4100
HTML version of Basic Foils prepared 15 March 1996

Foil 2 Abstract of CPS713-Case Study II
Initial NAS Benchmark Setup

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
This is first of three foilsets on CFD and NAS Benchmarks
This describes the four basic NAS benchmarks and their relation to the Navier Stokes Equations in the the 5 component CFD equations
We use opportunity to discuss time discretization and stepping in general based on Hirsch CFD book and Numerical Recipes
Stability, Implicit and Explicit formulations are introduced
Beam Warming Equations and their stability
HTML version of Basic Foils prepared 15 March 1996

Foil 3 NAS Parallel Benchmarks

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Here we summarize three foilsets CPS713NAS-I,II,III
This Section is Based on two publications
The NAS Parallel Benchmarks
  • David Bailey, John Barton, Thomas Lasinski,
  • Horst Simon NASA Ames
Sec. 3: A Methodology for Benchmarking some CFD Kernels on Highly Parallel Machines
  • Sisira Weeratunga, Eric Barszcz, Rod Fatoohi,
  • V. Venkatakrishnan
HTML version of Basic Foils prepared 15 March 1996

Foil 4 Summary of Sec 3: NAS Parallel Benchmarks:The CFD Kernels

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
A Collection of Iterative PDE Solvers embedded in a Pseudo CFD application program is proposed (and now used by community) for the performance evaluation of CFD Codes on highly parallel processors. The solvers are also now used as a general benchmark for different parallel machines. The pseudo application program is stripped of complexities associated with real CFD application programs, thereby enabling a simpler description of the algorithms. However it is capable of reproducing the essential computation and data access/movement characteristics of large scale state of the art CFD codes. These latter features make the CFD kernels suitable for CPS713 as a framework for discussing essential computer science issues in CFD.
HTML version of Basic Foils prepared 15 March 1996

Foil 5 Overview of Four Computational (PDE Solver) Kernels in
NAS Parallel Benchmarks

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
NASA analysed their implicit CFD codes and identified four computational kernels which typified the majority of three-dimensional Euler (non Viscous) and Navier-Stokes (Viscous) CFD programs using finite difference or finite volume methods on a structured grid.
HTML version of Basic Foils prepared 15 March 1996

Foil 6 First Two Computational Kernels

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
1)Solution of multiple independent systems of non diagonally dominant block tridiagonal equations with a 5 by 5 block size.
(i.e. a matrix made out of 5 by 5 blocks with one nonzero block on diagonal and one nonzero block either side of diagonal)
2)Solution of multiple independent systems of scalar (i.e. not block) pentadiagonal equations.
(i.e. matrix with nonzeros on diagonal and two nonzero elements either side of it.)
These two kernels are representative of computations associated with implicit operator in the NASA ARC3D code.
HTML version of Basic Foils prepared 15 March 1996

Foil 7 Second Two Computational Kernels

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
3) Regular Sparse 5 by 5 block matrix -vector multiplication
i.e. Sparse matrix-vector multiplication where each operation is multiplication by a 5 by 5 submatrix into a five dimensional subvector
  • This is typical of explicit part of almost all CFD solvers using structured grids.
4) Regular Sparse 5 by 5 block lower and upper triangular system solution
i.e same basic 5 by 5 submatrix into 5 dimensional vector operation with the arithmetic that of using LU decomposition on a sparse matrix.
  • This is typical of implicit solver used in newer codes such as INS3D-LU
HTML version of Basic Foils prepared 15 March 1996

Foil 8 The First Two Simplifications of the NAS Parallel Benchmarks

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
1)Absence of Realistic Boundary Conditions:
  • In a real CFD code,one would normally transform from a complicated physical domain into a computational domain where regular (structured) grid is appropriate. This transformation increases comptational complexity and storage requirements of real codes.
2)Simplified artificial dissipation
  • The benchmark adds a simple linear fourth order "artificial dissipation" term. In real codes an adaptive (dependent on pressure changes) mix of nonlinear second and fourth-difference terms are used. This simplification does not change computational structure significantly.
  • Artificial Dissipation is used to improve numerical stability of solver. The benchmark is designed to be very stable with more dissipation than normal.
HTML version of Basic Foils prepared 15 March 1996

Foil 9 The Second Two Simplifications of the NAS Parallel Benchmarks

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
3) Time Differencing Schemes
  • There are several clever time differencing schemes which have been developed to improve stability and accuracy of Navier Stokes Equations
    • Upwind Differencing
    • Flux-vector Splitting
    • Total Variation Diminishing -- TVD
    • Flux-difference Splitting
  • The benchmarks only use the combination of a particular implicit differencing with artificial dissipation
4) Absence of Turbulence Models
  • Forming the Reynolds Averaged Equations can introduce additional (typically one or two) differential equations and change values of parameters such as viscosity. This is absent in benchmark.
HTML version of Basic Foils prepared 15 March 1996

Foil 10 Basic Formalism for 5 Component CFD

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We can write the five basic variables of CFD as a five component vector given in Hirsch Chapters 16 and 22
In detail the five components are given by:
Note this uses Ideal Gas Equation of State P V = R T with V proportional to inverse of r
HTML version of Basic Foils prepared 15 March 1996

Foil 11 What is relation of NAS benchmarks to Navier Stokes Equations -- Terms E F G?

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
The terms E(U) F(U) and G(U) correspond to the non viscous terms in the Navier Stokes equations
  • Note that the enthalpy H = u(5) + pressure p and this appears in 5th component of E F and G
    • T is temperature
    • u(5) = r times E
    • Total energy E per unit mass = Internal Energy (cv T) plus external Energy (ux2 + uy2 + uz2)/2
    • Equation of State is for ideal gas
    • pressure p = r . T . constant(r) .
HTML version of Basic Foils prepared 15 March 1996

Foil 12 What is relation of NAS benchmarks to Navier Stokes Equations -- Terms T V W ?

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
The terms T (U,Ux) V(U,Uh) and W(U,Uz) include
  • A numerical viscosity term proportional to d(1..5)x,h,z which ends up if all the values of d are equal (which they almost are) as a equation like:
  • An approximation to Viscosity term in Navier Stokes Equation where ALL cross derivatives as listed below, are neglected
    • This approximation is implied by equation (3.1a) which doesn't allow such terms as those above.
    • Note also that numerical coefficient (k3 . k4) seems to be missing in 5th component of T U and W
  • The Ñ Temperature term for evolution of r E ( 5th componentof U)
HTML version of Basic Foils prepared 15 March 1996

Foil 13 What is relation of NAS benchmarks to Navier Stokes Equations-- Term H and Boundary Conditions ?

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
The term H is to viewed as an external force so arranged that a simple function exactly satisfies equations
The simple function is a linear combination of powers ( 0 through 4 ) of x h and z.
The boundary conditions and initial conditions are similarily set to have a very simple form
Note H does not depend on the unknown U in spite of what memo says!
H depends only on independent variables x h and z.
HTML version of Basic Foils prepared 15 March 1996

Foil 14 What is relation of NAS benchmarks to Navier Stokes Equations -- Space and Time Variables

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Note computational domain is the unit cube in x h and z space
We use these greek labels to indicate one has (notionally) transformed from original physical domain x h and z. (As noted the transformation matrix has been ignored)
We use t not t for two reasons
  • Firstly we probably have transformed time as well as space in map to computational domain
  • Secondly we only want steady state solution U and so we can view t as labelling iteration and not necessarily as time in a diffusive equation. This implies all that counts is accuracy at infinite time t and not accuracy at intermediate finite time t.
HTML version of Basic Foils prepared 15 March 1996

Foil 15 Treatment of Time Discretization for PDE's

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We will follow treatments in
  • Volume 1 of Numerical Computation of Internal and External Flows
  • Fundamentals of Numerical Discretization
  • C. Hirsch , Wiley 1988
  • And Numerical Recipes by
  • William Press, Saul Teukolsky, William Vetterling, and Brian Flannery
  • Cambridge University Press 1992
Note two key approachs:
  • Explicit: Values at next time explicitly calculable from those at previous time step(s)
  • Implicit: Values at next time given implicitly by matrix equations involving previous time step(s).
HTML version of Basic Foils prepared 15 March 1996

Foil 16 Treatment of Time Stepping for PDE's :
Numerical Recipes -- Chapter 19

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Equation (19.1.1) (19.1.6) are flux-conservative equations of general type of NAS benchmarks
Equation (19.1.11) is FTCS (Forward Time Centered Space) for (19.1.6)
Von Neumann Stability Analysis for constant coefficient equations
Lax Method Equation (19.1.14)
Courant Stability Condition Equation (19.1.17)
Physical Interpretation in Fig 19.1.3
Numerical Viscosity implied in Lax's Method Equation (19.1.19)
HTML version of Basic Foils prepared 15 March 1996

Foil 17 Comparison of Different Methods for Solving Linear Convection Equation

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
This discusses figures 8.3.6 to 8.3.8 of Hirsch
Convection Equation is (19.1.6) in Numerical Recipes
Lax-Friedrich's Scheme is called Lax Method -
Equations (19.1.14,15) in Numerical Recipes
First Order Upwind Scheme is Equation (19.1.27) and Figure 19.1.4 in Numerical Recipes
  • Note effect of numerical viscosity in Lax and Upwind methods which unphysically damps wave propagation in last two figures.
The Leapfrog Method is Figure 19.1.5 and
Equation 19.1.30 in Numerical Recipes
The Lax-Wendroff Method is Equations (19.1.37,38) and Figure 19.1.7 of Numerical Recipes
  • Both Leapfrog and Lax-Wendroff have formal errors which are second order in time step. Other two methods are only first order accurate.
  • The smaller numerical viscosity is good in figures 8.3.7,8 but causes some unphysical oscillations in figure 8.3.6 for both second order methods
HTML version of Basic Foils prepared 15 March 1996

Foil 18 Overview of Stability and Accuracy of Partial Differential Equation Solvers

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Note accuracy determines formally how fast the errors tend to zero as the time step d t tends to zero.
Stability determines if the errors build up or reinforce each other as one iterates in time
All time differencing schemes are valid representations of equations i.e.
reduce to differential equation in limit of dx and d t tend to zero
However not all differencing schemes are stable!
We can investigate stability for simple linear equations with von Neumann's Analysis
Let u(n)j be replaced by u(n)j + e(n)j where:
n labels time iteration and j spatial discretization
If the equations are linear in u, the e(n)j obeys same equations as u(n)j but with simple (zero) boundary conditions
If equation not linear, then this analysis still governs local behavior and gives a stability condition must must be satisfied by linearized equations
  • To linearize something like u ¶ (u) / ¶ t, replace varying u outside derivative by its local value.
HTML version of Basic Foils prepared 15 March 1996

Foil 19 Solution of Homogenous Equations by Fourier Analysis

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
If the Velocity v in convection equation is constant and
we have the natural boundary conditions for e
then Fourier Analysis diagonalizes difference equations
Note that general equation:
HTML version of Basic Foils prepared 15 March 1996

Foil 20 Solution of Linear Stability Equations for the Linear Convection Equation

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We expand error e in fashion suggested by Fourier solution
HTML version of Basic Foils prepared 15 March 1996

Foil 21 General Stability Analysis of some two-step Methods -- Basic Iteration Equation

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
See sec. 11.1 and table 11.1 of Hirsch, Volume 1.
This analysis introduced by Beam and Warming
(People not Hot Rods)
This is implicit unless b = 0 when it becomes explicit
This set of equations is defined in Equation (11.1.22) of Hirsch after he discusses a more general set involving K(n-1)
HTML version of Basic Foils prepared 15 March 1996

Foil 22 Differential Operator and its Eigenvalues for Beam Warming Equations

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Note we follow same notation as NAS Benchmark Paper Equation (3.11)
Hirsch calls our
For Stability Analysis write the linear form:
K = W U
where W is for one dimension and second order differencing a 5 by 5 matrix
When analysing the convection equation with K = -v ¶ / ¶x , we get W = -ivk is pure imaginary
HTML version of Basic Foils prepared 15 March 1996

Foil 23 Stability Conditions for Beam Warming Equations

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Note that Explicit Euler was always unstable for convection equation but for our case, the artificial viscosity ensures that explicit Euler is stable for small enough time step D t
Implicit Euler is always stable for dissipative systems because these by definition have Re(W) negative.
HTML version of Basic Foils prepared 15 March 1996

Foil 24 The effect of Nonlinearity on Implicit Equations

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Taking the special case q=0 and b=1 used in the NAS benchmark Equation (3.12)
Northeast Parallel Architectures Center, Syracuse University, npac@npac.syr.edu

If you have any comments about this server, send e-mail to webmaster@npac.syr.edu.

Page produced by wwwfoil on Sun Feb 22 1998