My goal is to develop and implement efficient, accurate, and robust Implicit-Explicit Runge-Kutta (IMEX RK) methods [9] for overcoming geometry-induced stiffness with applications to computational electromagnetics (CEM), computational fluid dynamics (CFD) and computational aeroacoustics (CAA). IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using implicit methods. Current algorithms in CEM can only simulate purely harmonic (up to lOGHz plane wave) EM scattering by fighter aircraft, which are assumed to be pure metallic shells, and cannot handle the inclusion of coatings, penetration into and radiation out of the aircraft. Efficient MEX RK methods could potentially increase current CEM capabilities by 1-2 orders of magnitude, allowing scientists and engineers to attack more challenging and realistic problems

Kanevsky, Alex

NASA Technical Reports Server

NASA GSRP Final Report (6/27/04) 
(1) Name and Grant Number: 
Alex Kanevsky, NGT-1-01024 
(2) Dates of Fellowship and Degrees Received: 
GSRP: 7/0 1/200 1 - 6/30/2004. 
Degree: M.Sc. in Applied Mathematics, 5/2001 
(3) Title: 
Overcoming Geometry-Induced Stiffness with IMplicit-Explicit (IMEX) Runge-Kutta 
Algorithms on Unstructured Grids with Applications to CEM, CFD, and CAA. 
(4) NASA Research Advisor: 
Dr. Mark H. Carpenter 
(5) Conference Participation: 
International Conference on Spectral and High Order Methods (ICOSAHOM), Brown 
University, Providence, RI, June 21-25, 2004. 
AFOSR Workshop on Advances and Challenges in Time-Integration of PDE's, Brown 
University, Providence, RI, August 18-20,2003. 
NASA GSRP Orientation and Workshop, Hampton, VA, July 2003. 
Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, 
June 17-20,2003. 
ICASE's 30th Anniversary Conference, Hampton, VA, August 2002. 
NASA GSRP Orientation and Workshop, Hampton, VA, July 24-26, 2002. 
SIAM 50th Anniversary and 2002 Annual Meeting, Philadelphia, PA, July 8-12,2002. 
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https://ntrs.nasa.gov/search.jsp?R=20050226977 2019-08-29T19:41:22+00:00Z
International Conference on Scientific Computing, Partial Differential Equations and 
Image Processing, On the Occasion of Stanley Osher’s 60h birthday, UCLA, April 5-7, 
2002. 
Ninth International Conference on Hyperbolic Problems: Theory, Numerics, 
Applications. California Institute of Technology, Pasadena, CA, March 25-29,2002. 
NASA GSRP Orientation and Workshop, Hampton, VA, July 25-27,2001 
(6) Seminars or Lectures: 
Seminar given at NASA GSRP Orientation and Workshop, Hampton, VA, July 2003. 
(7) Meetings Attended By Invitation: 
NIA 
Teaching Experience: 
(a) Instructor, Introduction to Computing Sciences (AM 16), Brown University, 
Spring 2004. 
(b) Teaching Assistant, Methods in Applied Math I (AM 33), Brown University, Fall 
2002. 
Summary of Research: See Attached Page. 
(10) Publications and Papers: 
Chauviere, C., Hesthaven, J.S., Kanevsky, A., and Warburton, T., High-order localized 
time integration for grid-induced stifiess, Proceedings of the Second MIT conference on 
Computational Fluid and Solid Mechanics (June 2003). 
(11) Patents: 
NIA 
(12) Current Forwarding Address: 
2 
Alex Kanevsky 
Division of Applied Mathematics 
Brown University 
182 George Street 
Providence, RI 02912 
(13) Appraisal of the GSRP: 
The GSRP program has been incredibly valuable for my development as a 
scientisthesearcher. I would like to thank all those individuals at NASA from the bottom 
of my heart who gave me such a wonderful opportunity. The most valuable aspect of the 
program was the mentoring I received from Dr. Mark H. Carpenter. Mark is a scientist of 
the highest caliber with incredible clarity, vision and creativity. He is also a very nice 
man. He really taught me a lot during all the times I came down during the summers and 
winters. I learned about the smaller and bigger pictures, including how to break down and 
analyze problems mathematically as well as what types of practical uses and applications 
IMEX methods have in science and engineering. I would like to thank him for all of these 
things. 
I gained a lot from the annual GSRP conferences organized by Lloyd Evans and David 
Peterson. It was very beneficial to meet other GSRP participants and learn about their 
research. I also enjoyed the tours of NASA and all of the social activities. Thank you! 
FACULTY ADVISER: J 9 Lh 
STUDENT: DATE: 
3 
L 
(9) Summary of Research: 
My goal is to develop and implement efficient, accurate, and robust IMplicit-Explicit Runge- 
Kutta (IMEX RK) methods [9] for overcoming geometry-induced stiffness with applications to 
computational electromagnetics (CEM), computational fluid dynamics (CFD) and computational 
aeroacoustics (CAA). IMEX algorithms solve the non-stiff portions of the domain using explicit 
methods, and isolate and solve the more expensive stiff portions using implicit methods. 
Current algorithms in CEM can only simulate purely harmonic (up to lOGHz plane wave) EM 
scattering by fighter aircraft, which are assumed to be pure metallic shells, and cannot handle the 
inclusion of coatings, penetration into and radiation out of the aircraft. Efficient MEX RK 
methods could potentially increase current CEM capabilities by 1-2 orders of magnitude, 
allowing scientists and engineers to attack more challenging and realistic problems. 
This year, I completed my third year of research under the guidance of Professors David Gottlieb 
and Jan S .  Hesthaven of Brown University and Dr. Mark H. Carpenter of NASA Langley 
Research Center. During the past 3 years, I implemented and tested explicit, implicit, and IMEX 
time-integration algorithms for solving linear as well as non!inear equations in one and two 
dimensions on unstructured grids, such as burgers equation and the Euler equations, using 
Discontinuous Galerkin [3,4,5,8] spectral element spatial discretizations. 
The Discontinuous Galerkin Spectral Element (DGSE) method builds upon the strengths and 
overcomes the weaknesses of the Discontinuous Galerkin Finite Element method [3,4] and the 
classical spectral element method introduced by Patera [I 1, 121, and has a number of advantages 
over classical finite difference and finite volume methods. DGSE methods are especially well 
suited for IMEX algorithms, since they allow for clean and easy decoupling of the stiff from the 
nonstiff regions of the domain. Furthermore, they are highly parallelizable and accurate, provide 
for simple treatment of boundary conditions, handle complicated geometries well, and can easily 
handle adaptivity. Utilizing an unstructured grid [8] allows me to capture very fine details of 
complex domains, which is crucial for tackling real-world problems. 
My research shows that when geometry-induced stiffness is significant (greater or equal to 2 
orders of magnitude), IMEX algorithms [9] outperform traditional explicit time-stepping RK 
methods by about an order of magnitude or more in 1D for systems having smooth solutions. I 
also found that Krylov subspace iterative methods, such as GMRES and BiCGStab, in 
conjunction with good preconditioners are essential in order to achieve this kind of speedup. 
Some of the results from my research have been presented at the Second MIT conference on 
Computational Fluid and Solid Mechanics and are published in [2]. 
In 2D, IMEX methods outperform explicit methods even without preconditioning (when 
geometry-induced stiffness is greater than 1-2 orders of magnitude) for problems having smooth 
solutions. When shocks are present, the situation is not as clear since Newton’s method often 
fails to converge and many steps must be repeated with smaller time-steps. Preconditioning gives 
IMEX methods an even greater computational advantage in two dimensions. ILU 
4 
preconditioners worked very well in lD, while Jacobi, Gauss-Seidel and block-Jacobi 
preconditioners were not as effective. I will need to further investigate effective 2D and 3D 
preconditioning techniques such as multignd-based preconditioners [ 101. 
I am currently developing and testing time advancement algorithms for solving systems of 
nonlinear partial differential equations in multiple dimensions, such as Eulers equations and the 
compressible Navier-Stokes equations, using Discontinuous Galerkin [8] spectral element spatial 
discretizations. 
2D Nozzle Flow and Conclusion: 
The three-dimensional Navier-Stokes (NS) equations describe the behavior of many types of 
fluids and are given below: 
L 
Re = = Reynoldsnumber, 
+ 
P = dynamic viscosity, A. = bulk viscosity, k = coefficient of thermal conductivity, 
( if' :; ) ' d ,  E?--, 6,, = Kronecker delta, stress tensorelements T. I = /I  7 + 7 + o I 
C?. 
t heal capacity (constant pressure 
= heat capacity (conslantvolume) 
I' = 
For our test case, we solve the two-dimensional Euler equations (in a nozzle), which can be 
derived from the NS equations by taking the limit of Reynolds number going to infinity (right- 
hand side in above equations becomes 0). 
5 
The axjsymmetric converging-diverging nozzle has area given by: 
Area(x) = 1.75 - .75 * cos( (.2 * x- 1.0) * pi), 0 5 x 5 5.0 
Area(x) = 1.25 - .25 * cos( (.2 * x - 1.0) * pi ), 5.0 s x 5 10.0 
We solve a steady-state (SS) problem where the SS solution has a shock at x = 7.56. The inflow 
Mach number = ,240 and the outflow Mach number = 501. The test case until the physical time t 
= 5, which is well before the time the shock develops. So h r  our test, the SoMon is relatively 
smooth and does not contain shocks, although it does contain some complex structures. 
Densilv x-Momenbm 
1 1 
0 5  0.1 
n n - -
0 2 4 6 8 10 0 2 4 6 8 10 
y-Momentum Mach Number 
0 2 4 6 8 10 0 2 4 6 E 10 
Rho-Residual 
J I 
0 1 2 3 4  5 6  0 1 2 3 4  5 6 
Figure 1: IMEX, N-8, No Preconditioning, Newton Tol. - le-3, Bicgrtab. Failed-0. nozzk52, 
CPU-time-11,115.(5.194timestep.) 
The implicit elements are located within a semicircular region of radius r = .25 centered at x = 
7.562 (center lies on bottom wall/centerlioe, see figure). The geometric stifhess, which is 
deiined as the ratio of average IMEX time step to average explicit time step is approximately 
2.27 (N = 8) fix the unstructured triangular mesh used. The ratio of implicit to total elements is 
56/496, which is slightly greater than 10%. 
We use the method of lines to discrethe the partial differential equations m space and time. First, 
we discretize space usmg a nodal Discontinuous Galerkin Spectral Element method based on [8]. 
Next, we integrate the system of ordinary difkentbl equations (ODES) m time. For the IMEX 
method, we use a modified Newton-Krybv method (Newton tolerance = 1E-03) to linearize the 
6 
nonlinear system of ODES, and to iteratively solve the linear systems (BiCGStab). Both time- 
integration methods are globally 4'h-order accurate. Preconditioners were not used. 
Time- 
Integration 
We impose boundary conditions on the characteristic variables at the inflow and outflow, and 
penalize the velocity against its mirror image at the top and bottom walls. 
Polynomial Average Total Time- 
Order (N) Time-Step Steps 
We use a stability time-step controller for all runs: 
Method 
Purely Explicit 
At = CFU(h*N**geometric factor), where h is the maximum wave speed. 
' (At) - - 
4 1.39E-03 3,601 
We can see that for N = 8 both methods take approximately the same CPU-time, with the IMEX 
being slightly faster. For N = 4, the explicit method is faster. 
lMEX 
8 4.24E-04 1 1,800 
4 3.34E-03 1,499 
8 9.63E-04 5,194 
time (sec.) 
1,349 
11,313 
11,115  
IMEX methods are not faster in this particular test, but will be significantly faster for the 
following situation: 
(1) 
(2) Effective Preconditioners. 
(3) Geometric Stiffness > 2. 
(4) 
(5) Advanced Time-Step Controller (PID). 
Ratio of implicit to total elements e 5-10%. 
Navier Stokes equations, which have viscous terms. 
We believe that when some or all of the above are implemented, IMEX methods will become 
significantly more efficient than purely explicit time-integration methods. 
7 
References 
[I] J.C. Butcher, The Numerical Analysis of Ordinary DifSerential Equations, Wiley, 1987. 
[2] C. Chauviere, J. S. Hesthaven, A. Kanevsky, and T. Warburton, High-order localized time 
integration for grid-induced stifiess, Proceedings of the Second MIT conference on 
Computational Fluid and Solid Mechanics (June 2003). 
[3] B. Cockburn, An introduction to the discontinuous Galerkin methodfor convection 
dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic 
Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), 
Lecture Notes in Mathematics 1697, Springer-Verlag, pp. 15 1 -268( 1998). 
[4] B. Cockburn, G. E. Karniadakis and C.-W. Shu (Editors), Discontinuous Galerkin 
Methods: Theory, Computation and Applications. Lecture Notes in Computational Science 
and Engineering 1 1, Springer-Verlag, 2000. 
[5] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection- 
clcnzimted problems, J. Sci. Comput. 16(3), pp. 173-261 (2001). 
[6] E. Hairer, S. P. Ngrsett and G. Wanner, Solving Ordinary Differential Equations I :  NonstifS 
Problems, Springer-Verlag, 1987. 
171 E. Hairer, G .  Wanner, Solving Ordinary Differential Equations II: Stiff and Differential 
Algebraic Problems, Springer-Verlag, 1991. 
[8] J. S. Hesthaven and T. Warburton, Nodal High-Order Methods on Unstructured Grids I .  
Time-Domain Solution of Maxwell’s Equations, J. Comput. Phys. 181, pp. 186-221 (2002). 
[9] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection- 
difSusion-reaction equations, Appl. Numer. Math. 44, pp. 139-1 8 1 (2003). 
[ 101 D. A. Knoll and D. E. Keyes, Jacobian-free Newton Krylov methods: a survey of 
approaches and applications, J. Comput. Phys. 193, pp. 357-397 (2004). 
[ 111 K. Z.  Korczak and A. T. Patera, An Isoparametric Spectral Element Method for Solution of 
the Navier-Stokes Equations in Complex Geometries, J. Comput. Phys. 62, pp. 36 1-382 
(1986). 
[ 121 A. T. Patera, A Spectral Element Method for Fluid Mechanics: Laminar Flow in a 
Channel Expansion, J. Comput. Phys. 54, pp. 468-488(1984). 
[ 131 Y. Saad, Iterative Methods for  Sparse Linear Systems, PWS Publishing Co., 1996. 
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Overcoming Geometry-Induced Stiffness with IMplicit-Explicit (IMEX) Runge-Kutta Algorithms on Unstructured Grids with Applications to CEM, CFD, and CAA

Overcoming Geometry-Induced Stiffness with IMplicit-Explicit (IMEX) Runge-Kutta Algorithms on Unstructured Grids with Applications to CEM, CFD, and CAA

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