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Rosenbrock-Wanner Methods: Construction and Mission Kapitel lesen Erstes Kapitel lesen

2021 | OriginalPaper | Buchkapitel

Exponential Rosenbrock Methods and Their Application in Visual Computing

verfasst von : Vu Thai Luan, Dominik L. Michels

Erschienen in: Rosenbrock—Wanner–Type Methods

Verlag: Springer International Publishing

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Abstract

We introduce a class of explicit exponential Rosenbrock methods for the time integration of large systems of stiff differential equations. Their application with respect to simulation tasks in the field of visual computing is discussed where these time integrators have shown to be very competitive compared to standard techniques. In particular, we address the simulation of elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects like stability and energy conservation, large stiffnesses, high fidelity and visual accuracy.

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Vorheriges Kapitel Water and Hydrogen Flow in Networks: Modelling and Numerical Solution by ROW Methods
Nächstes Kapitel W-Methods and Approximate Matrix Factorization for Parabolic PDEs with Mixed Derivative Terms
Fußnoten
1
We estimated the error after 60 s of simulated time based on the accumulated Euclidean distances of the individual particles in the position space compared to ground truth values which are computed with a sufficiently small step size.
 
2
In order to detect collisions efficiently, we make use of a standard bounding volume hierarchy.
 
Literatur
1.
A.H. Al-Mohy, N.J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011) MathSciNetCrossRef
2.
U. Ascher, S. Ruuth, B. Wetton, Implicit-explicit methods for time-dependent PDEs. SIAM J. Numer. Anal. 32(3), 797–823 (1997) CrossRef
3.
D. Baraff, A. Witkin, Large steps in cloth simulation, in ACM Transactions on Graphics, SIGGRAPH’98 (ACM, New York, 1998), pp. 43–54
4.
M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, E. Grinspun, Discrete elastic rods. ACM Trans. Graph. 27(3), 63:1–63:12 (2008)
5.
M. Caliari, A. Ostermann, Implementation of exponential Rosenbrock-type integrators. Appl. Numer. Math. 59(3–4), 568–581 (2009) MathSciNetCrossRef
6.
M. Caliari, P. Kandolf, A. Ostermann, S. Rainer, The Leja method revisited: backward error analysis for the matrix exponential. SIAM J. Sci. Comput. 38(3), A1639–A1661 (2016) MathSciNetCrossRef
7.
W.L. Chao, J. Solomon, D. Michels, F. Sha, Exponential integration for Hamiltonian Monte Carlo, in Proceedings of the 32nd International Conference on Machine Learning, ed. by F. Bach, D. Blei. Proceedings of Machine Learning Research, vol. 37, pp. 1142–1151 (PMLR, Lille, 2015)
8.
Y.J. Chen, U.M. Ascher, D.K. Pai, Exponential Rosenbrock-Euler integrators for elastodynamic simulation. IEEE Trans. Visual. Comput. Graph. 24(10), 2702–2713 (2018) CrossRef
9.
S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430–455 (2002) MathSciNetCrossRef
10.
C. Curtiss, J.O. Hirschfelder, Integration of stiff equations. Proc. Natl. Acad. Sci. 38(3), 235–243 (1952) MathSciNetCrossRef
11.
C.F. Curtiss, J.O. Hirschfelder, Integration of stiff equations. Proc. Natl. Acad. Sci. USA 38(3), 235–243 (1952) MathSciNetCrossRef
12.
B. Eberhardt, O. Etzmuß, M. Hauth, Implicit-explicit schemes for fast animation with particle systems, in Proceedings of the 11th Eurographics Workshop on Computer Animation and Simulation (EGCAS) (Springer, Berlin, 2000), pp. 137–151
13.
S. Gaudreault, J. Pudykiewicz, An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere. J. Comput. Phys. 322, 827–848 (2016) MathSciNetCrossRef
14.
C. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice–Hall, Englewood Cliffs, 1971)
15.
S. Geiger, G. Lord, A. Tambue, Exponential time integrators for stochastic partial differential equations in 3D reservoir simulation. Comput. Geosci. 16(2), 323–334 (2012) CrossRef
16.
R. Goldenthal, D. Harmon, R. Fattal, M. Bercovier, E. Grinspun, Efficient simulation of inextensible cloth, in ACM Transactions on Graphics, SIGGRAPH’07 (2007)
17.
M.A. Gondal, Exponential Rosenbrock integrators for option pricing. J. Comput. Appl. Math. 234(4), 1153–1160 (2010) MathSciNetCrossRef
18.
E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, New York, 1996) CrossRef
19.
M. Hauth, O. Etzmuss, A high performance solver for the animation of deformable objects using advanced numerical methods. Comput. Graph. Forum 20, 319–328 (2001) CrossRef
20.
N.J. Higham, Functions of Matrices: Theory and Computation (SIAM, Philadelphia, 2008) CrossRef
21.
M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997) MathSciNetCrossRef
22.
M. Hochbruck, A. Ostermann, Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2005) MathSciNetCrossRef
23.
M. Hochbruck, A. Ostermann, Explicit integrators of Rosenbrock-type. Oberwolfach Rep. 3, 1107–1110 (2006)
24.
25.
M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998) MathSciNetCrossRef
26.
M. Hochbruck, A. Ostermann, J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009) MathSciNetCrossRef
27.
28.
M.W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901) MATH
29.
V.T. Luan, Fourth-order two-stage explicit exponential integrators for time-dependent PDEs. Appl. Numer. Math. 112, 91–103 (2017) MathSciNetCrossRef
30.
31.
V.T. Luan, A. Ostermann, Explicit exponential Runge–Kutta methods of high order for parabolic problems. J. Comput. Appl. Math. 256, 168–179 (2014) MathSciNetCrossRef
32.
V.T. Luan, A. Ostermann, Exponential Rosenbrock methods of order five–construction, analysis and numerical comparisons. J. Comput. Appl. Math. 255, 417–431 (2014) MathSciNetCrossRef
33.
V.T. Luan, A. Ostermann, Stiff order conditions for exponential Runge–Kutta methods of order five, in Modeling, Simulation and Optimization of Complex Processes - HPSC 2012, H.B. et al. (ed.) (Springer, Berlin, 2014), pp. 133–143
34.
V.T. Luan, A. Ostermann, Parallel exponential Rosenbrock methods. Comput. Math. Appl. 71, 1137–1150 (2016) MathSciNetCrossRef
35.
V.T. Luan, J.A. Pudykiewicz, D.R. Reynolds, Further development of the efficient and accurate time integration schemes for meteorological models J. Comput. Sci. 376, 817–837 (2018) MathSciNetMATH
36.
D.L. Michels, M. Desbrun, A semi-analytical approach to molecular dynamics. J. Comput. Phys. 303, 336–354 (2015) MathSciNetCrossRef
37.
D.L. Michels, J.P.T. Mueller, Discrete computational mechanics for stiff phenomena, in SIGGRAPH ASIA 2016 Courses (2016), pp. 13:1–13:9
38.
D.L. Michels, G.A. Sobottka, A.G. Weber, Exponential integrators for stiff elastodynamic problems. ACM Trans. Graph. 33(1), 7:1–7:20 (2014)
39.
D.L. Michels, J.P.T. Mueller, G.A. Sobottka, A physically based approach to the accurate simulation of stiff fibers and stiff fiber meshes. Comput. Graph. 53B, 136–146 (2015) CrossRef
40.
D.L. Michels, V.T. Luan, M. Tokman, A stiffly accurate integrator for elastodynamic problems. ACM Trans. Graph. 36(4), 116 (2017)
41.
J. Niesen, W.M. Wright, Algorithm 919: a Krylov subspace algorithm for evaluating the φ-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38, 3 (2012) MathSciNetCrossRef
42.
D.A. Pope, An exponential method of numerical integration of ordinary differential equations. Commun. ACM 6, 491–493 (1963) MathSciNetCrossRef
43.
44.
A. Stern, M. Desbrun, Discrete geometric mechanics for variational time integrators, in SIGGRAPH 2006 Courses (2006), pp. 75–80
45.
A. Stern, E. Grinspun, Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul. 7, 1779–1794 (2009) MathSciNetCrossRef
46.
A. Tambue, I. Berre, J.M. Nordbotten, Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods. Adv. Water Resour. 53, 250–262 (2013) CrossRef
47.
D. Terzopoulos, J. Platt, A. Barr, K. Fleischer, Elastically deformable models. ACM Trans. Graph. 21, 205–214 (1987)
48.
M. Tokman, J. Loffeld, P. Tranquilli, New adaptive exponential propagation iterative methods of Runge-Kutta type. SIAM J. Sci. Comput. 34, A2650–A2669 (2012) MathSciNetCrossRef
49.
H. Zhuang, I. Kang, X. Wang, J.H. Lin, C.K. Cheng, Dynamic analysis of power delivery network with nonlinear components using matrix exponential method, in 2015 IEEE Symposium on Electromagnetic Compatibility and Signal Integrity (IEEE, Piscataway, 2015), pp. 248–252
Metadaten
Titel
Exponential Rosenbrock Methods and Their Application in Visual Computing
verfasst von
Vu Thai Luan
Dominik L. Michels
Copyright-Jahr
2021
Buch
Rosenbrock—Wanner–Type Methods

Print ISBN: 978-3-030-76809-6

Electronic ISBN: 978-3-030-76810-2

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-76810-2_3

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