The time-discontinuous Galerkin (TDG) method possesses high-order accuracy and desirable C-and L-stability for second-order hyperbolic systems including structural acoustics. C- and L-stability provide asymptotic annihilation of high frequency response due to spurious resolution of small scales. These non-physical responses are due to limitations in spatial discretization level for large-complex systems. In order to retain the high-order accuracy of the parent TDG method for high temporal approximation orders within an efficient multi-pass iterative solution algorithm which maintains stability, generalized gradients of residuals of the equations of motion expressed in state-space form are added to the TDG variational formulation. The resultant algorithm is shown to belong to a family of Pade approximations for the exponential solution to the spatially discrete hyperbolic equation system. The final form of the algorithm uses only a few iteration passes to reach the order of accuracy of the parent solution. Analysis of the multi-pass algorithm shows that the first iteration pass belongs to the family of (p+1)-stage stiff accurate Singly-Diagonal-Implicit-Runge-Kutta (SDIRK) method. The methods developed can be viewed as a generalization to the SDIRK method, retaining the desirable features of efficiency and stability, now extended to high-order accuracy. An example of a transient solution to the scalar wave equation demonstrates the efficiency and accuracy of the multi-pass algorithms over standard second-order accurate single-step/single-solve (SS/SS) methods.

1.
Hughes, T. J., 2000. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publication, Inc.
2.
Dahlquist
G. G.
,
1963
. “
A special stability problem for linear multistep methods
”.
BIT
,
3
[], pp.
27
43
.
3.
Chung
J.
, and
Hulbert
G. M.
,
1993
. “
A time integration algorithm for structural dynamics with improved numerical dissipation: The Generalized-α method
”.
Journal of Applied Mechanics, TRANS. ASME
,
60
[], pp.
371
375
.
4.
Hilber
H. M.
,
Hughes
T. J. R.
, and
Taylor
R. L.
,
1977
. “
Improved numerical dissipation for time integration algorithms in structural dynamics
”.
Earthquake Engineering and Structural Dynamics
,
5
[], pp.
283
292
.
5.
Wood
W. L.
,
Bossak
M.
, and
Zienkiewicz
O. C.
,
1981
. “
An alpha modification of Newmark’s method
”.
International Journal for Numerical Methods in Engineering
,
15
[], pp.
1562
1566
.
6.
Austin
M.
,
1993
. “
High-order integration of smooth dynamical systems: Theory and numerical experiments
”.
International Journal for Numerical Methods in Engineering
,
36
[], pp.
2107
2122
.
7.
Hulbert
G. M.
,
1992
. “
Time finite element methods for structural dynamics
”.
International Journal for Numerical Methods in Engineering
,
33
[], pp.
307
331
.
8.
Thompson
L. L.
, and
He
D.
,
2005
. “
Adaptive space-time finite element methods for the wave equation on unbounded domains
”.
Computer Methods in Applied Mechanics and Engineering
,
194
[], pp.
1947
2000
.
9.
Subbarayalu, S., and Thompson, L. L., 2004. hp-adaptive time-discontinuous Galerkin finite element methods for time-dependent waves. Proceedings of 2004 ASME International Mechanical Engineering Congress & Exposition, Nov. 2004, Anaheim, CA., Paper IMECE 2004-60403.
10.
Kunthong
P.
, and
Thompson
L. L.
,
2005
. “
An efficient solver for the high-order accurate time-discontinuous galerkin (TDG) method for second-order hyperbolic systems
”.
Finite Elements in Analysis & Design
,
141
(
7–8)
[], pp.
729
762
.
11.
Kunthong, P., 2005. Efficient Predictor/Multi-corrector algorithms based on high-order accurate time-discontinuous Galerkin methods for second-order Hyperbolic systems. PhD thesis, Clemson University, Department of Civil Engineering, Clemson, South Carolina.
12.
Franca
L. P.
, and
Carmo
E. G. D. D.
,
1989
. “
The Galerkin gradient least-squares method
”.
Computer Methods in Applied Mechanics and Engineering
,
74
[], pp.
41
54
.
13.
Thompson, L. L., and Kunthong, P., 2005. “A residual based variational method for reducing dispersion error in finite element methods”. In Paper IMECE2005-80551, Proceedings IMECE’05, The American Society of Mechanical Engineers. 2005 International Mechanical Engineering Conference & Exposition by the Noise Control & Acoustics Division.
14.
Thompson
L. L.
, and
Sankar
S.
,
2001
. “
Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner-Mindlin plates
”.
Int. J. Numer. Meth. Engng
,
50
(
11)
[], pp.
2521
2545
.
15.
Thompson
L. L.
, and
Pinsky
P. M.
,
1996
. “
A space-time finite element method for structural acoustics in infinite domains, Part I: Formulation, stability, and convergence
”.
Comput. Methods in Appl. Mech. Engrg
,
132
[], pp.
195
227
.
16.
Hulbert, G. M., 1989. Space-Time Finite Element Methods for Second-Order Hyperbolic Equations. PhD thesis, Stanford University, Stanford, California.
17.
Baker, Jr., G. A., 1975. Essesntials of Pade´ Approximants. Academic Press, Inc., 111 Fifth Avenue, New York, New York 10003.
18.
Hulbert
G. M.
,
1994
. “
A unified set of single-step asymptotic annihilation algorithms for structural dynamics
”.
Computer Methods in Applied Mechanics and Engineering.
,
113
[], pp.
1
9
.
19.
Burington, R. S., 1972. Handbook of Mathematical Table and Formulas, fifth ed. McGraw-Hill Book Company.
20.
Rainville, E. D., 1960. Special Functions. The Macmillan Company, New York, USA.
21.
Hairer, E., and Wanner, G., 1991. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin.
22.
O´Neil, P. V., 1995. Advanced Engineering Mathematics, forth ed. Brooks/Cole Publishing Company, Pacific Grove, CA, USA.
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