August  2020, 40(8): 4597-4624. doi: 10.3934/dcds.2020194

Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise

1. 

Department of Computer science, Electrical engineering and Mathematical sciences, Western Norway University of Applied Sciences, Inndalsveien 28, 5063 Bergen, Norway, Center for Research in Computational and Applied Mechanics (CERECAM)

2. 

Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa, The African Institute for Mathematical Sciences(AIMS) of South Africa

3. 

Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany

* Corresponding author: Antoine Tambue, email: antonio@aims.ac.za

Received  January 2019 Revised  February 2020 Published  May 2020

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $ L^2 $ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $ \mathcal{O}\left(h^2\left(1+\max(0, \ln\left(t_m/h^2\right)\right)\right. \left.+\Delta t^{\frac{1}{2}}\right) $. Numerical simulations to illustrate our theoretical finding are provided.

Citation: Antoine Tambue, Jean Daniel Mukam. Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4597-4624. doi: 10.3934/dcds.2020194
References:
[1]

S. BlanesF. CasasJ. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[2]

S. BlanesF. CasasJ. A. Oteo and J. Ros, Magnus and Fer expansion for matrix differential equations: The convergence problem, J. Phys. A., 31 (1998), 259-268.  doi: 10.1088/0305-4470/31/1/023.  Google Scholar

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S. Blanes and P. C. Moan, Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems, Appl. Numer. Math., 56 (2006), 1519-1537.  doi: 10.1016/j.apnum.2005.11.004.  Google Scholar

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C. GonzálezA. Ostermann and M. Thalhmmer, A second-order Magnus-type integrator for nonautonomous parabolic problems, J. Comput. Appl. Math., 189 (2006), 142-156.  doi: 10.1016/j.cam.2005.04.036.  Google Scholar

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C. González and A. Ostermann, Optimal convergence results for Runge-Kutta discretizations of linear nonautonomous parabolic problems, BIT, 39 (1999), 79-95.  doi: 10.1023/A:1022369208270.  Google Scholar

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E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

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D. HippM. Hochbruck and A. Ostermann, An exponential integrator for non-autonomous parabolic problems, Electron. Trans. Numer. Anal., 41 (2014), 497-511.   Google Scholar

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M. Hochbruck and C. Lubich, On Magnus integrators for time-dependent Schrödinger equations, SIAM. J. Numer. Anal., 41 (2003), 945-963.  doi: 10.1137/S0036142902403875.  Google Scholar

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A. JentzenP. Kloeden and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950.  doi: 10.1214/10-AAP711.  Google Scholar

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A. Jentzen and M. Röckner, Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise, J. Differential Equations, 252 (2012), 114-136.  doi: 10.1016/j.jde.2011.08.050.  Google Scholar

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M. KovácsS. Larsson and F. Lindgren, Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise, Numer. Algorithms, 53 (2010), 309-220.  doi: 10.1007/s11075-009-9281-4.  Google Scholar

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R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055.  Google Scholar

[15]

S. Larsson, Nonsmooth data error estimates with applications to the study of the long-time behavior of the finite elements solutions of semilinear parabolic problems, Chalmers University of Technology, 1992. Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.1250 Google Scholar

[16]

G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2012), 515-543.  doi: 10.1093/imanum/drr059.  Google Scholar

[17]

G. J. Lord and A. Tambue, A modified semi-implict Euler-Maruyama scheme for finite element discretization of SPDEs with additive noise, Appl. Math. Comput., 332 (2018), 105-122.  doi: 10.1016/j.amc.2018.03.014.  Google Scholar

[18]

Y. Y. Lu, A fourth-order Magnus scheme for Helmholtz equation, J. Compt. Appl. Math., 173 (2005), 247-253.  doi: 10.1016/j.cam.2004.03.010.  Google Scholar

[19]

M. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

[20]

J. D. Mukam and A. Tambue, Strong convergence analysis of the stochastic exponential Rosenbrock scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise, J. Sci. Comput., 74 (2018), 937-978.  doi: 10.1007/s10915-017-0475-y.  Google Scholar

[21]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.  Google Scholar

[24]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 1993.  Google Scholar

[25]

J. Seidler, Da Prato-Zabczyk's maximal inequality revisited. I, Math. Bohem., 118 (1993), 67-106.   Google Scholar

[26]

A. Tambue and J. D. Mukam, Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non autonomous parabolic PDE with nonsmooth initial data, preprint, arXiv: 1809.03227v1. Google Scholar

[27]

X. Wang, Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise, IMA J. Numer. Anal., 37 (2017), 965-984.  doi: 10.1093/imanum/drw016.  Google Scholar

[28]

X. Wang and Q. Ruisheng, A note on an accelerated exponential Euler method for parabolic SPDEs with additive noise, Appl. Math. Lett., 46 (2015), 31-37.  doi: 10.1016/j.aml.2015.02.001.  Google Scholar

[29]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384.  doi: 10.1137/040605278.  Google Scholar

show all references

References:
[1]

S. BlanesF. CasasJ. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[2]

S. BlanesF. CasasJ. A. Oteo and J. Ros, Magnus and Fer expansion for matrix differential equations: The convergence problem, J. Phys. A., 31 (1998), 259-268.  doi: 10.1088/0305-4470/31/1/023.  Google Scholar

[3]

S. Blanes and P. C. Moan, Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems, Appl. Numer. Math., 56 (2006), 1519-1537.  doi: 10.1016/j.apnum.2005.11.004.  Google Scholar

[4]

H. Fujita and T. Suzuki, Evolutions problems, in Handbook of Numerical Analysis, Handb. Numer. Anal., 2, North-Holland, Amsterdam, 1991,789–928. doi: Evolutionproblems.  Google Scholar

[5]

C. GonzálezA. Ostermann and M. Thalhmmer, A second-order Magnus-type integrator for nonautonomous parabolic problems, J. Comput. Appl. Math., 189 (2006), 142-156.  doi: 10.1016/j.cam.2005.04.036.  Google Scholar

[6]

C. González and A. Ostermann, Optimal convergence results for Runge-Kutta discretizations of linear nonautonomous parabolic problems, BIT, 39 (1999), 79-95.  doi: 10.1023/A:1022369208270.  Google Scholar

[7]

E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[9]

D. HippM. Hochbruck and A. Ostermann, An exponential integrator for non-autonomous parabolic problems, Electron. Trans. Numer. Anal., 41 (2014), 497-511.   Google Scholar

[10]

M. Hochbruck and C. Lubich, On Magnus integrators for time-dependent Schrödinger equations, SIAM. J. Numer. Anal., 41 (2003), 945-963.  doi: 10.1137/S0036142902403875.  Google Scholar

[11]

A. JentzenP. Kloeden and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950.  doi: 10.1214/10-AAP711.  Google Scholar

[12]

A. Jentzen and M. Röckner, Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise, J. Differential Equations, 252 (2012), 114-136.  doi: 10.1016/j.jde.2011.08.050.  Google Scholar

[13]

M. KovácsS. Larsson and F. Lindgren, Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise, Numer. Algorithms, 53 (2010), 309-220.  doi: 10.1007/s11075-009-9281-4.  Google Scholar

[14]

R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055.  Google Scholar

[15]

S. Larsson, Nonsmooth data error estimates with applications to the study of the long-time behavior of the finite elements solutions of semilinear parabolic problems, Chalmers University of Technology, 1992. Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.1250 Google Scholar

[16]

G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2012), 515-543.  doi: 10.1093/imanum/drr059.  Google Scholar

[17]

G. J. Lord and A. Tambue, A modified semi-implict Euler-Maruyama scheme for finite element discretization of SPDEs with additive noise, Appl. Math. Comput., 332 (2018), 105-122.  doi: 10.1016/j.amc.2018.03.014.  Google Scholar

[18]

Y. Y. Lu, A fourth-order Magnus scheme for Helmholtz equation, J. Compt. Appl. Math., 173 (2005), 247-253.  doi: 10.1016/j.cam.2004.03.010.  Google Scholar

[19]

M. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

[20]

J. D. Mukam and A. Tambue, Strong convergence analysis of the stochastic exponential Rosenbrock scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise, J. Sci. Comput., 74 (2018), 937-978.  doi: 10.1007/s10915-017-0475-y.  Google Scholar

[21]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.  Google Scholar

[24]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 1993.  Google Scholar

[25]

J. Seidler, Da Prato-Zabczyk's maximal inequality revisited. I, Math. Bohem., 118 (1993), 67-106.   Google Scholar

[26]

A. Tambue and J. D. Mukam, Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non autonomous parabolic PDE with nonsmooth initial data, preprint, arXiv: 1809.03227v1. Google Scholar

[27]

X. Wang, Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise, IMA J. Numer. Anal., 37 (2017), 965-984.  doi: 10.1093/imanum/drw016.  Google Scholar

[28]

X. Wang and Q. Ruisheng, A note on an accelerated exponential Euler method for parabolic SPDEs with additive noise, Appl. Math. Lett., 46 (2015), 31-37.  doi: 10.1016/j.aml.2015.02.001.  Google Scholar

[29]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384.  doi: 10.1137/040605278.  Google Scholar

Figure 1.  Convergence of the Magnus integrator for $ \beta = 1 $, and $ \beta = 2 $ in (154). The order of convergence in time is $ 0.57 $ for $ \beta = 1 $, $ 0.54 $ for $ \beta = 2 $. The total number of samples used is $ 100 $
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