Peer methods for ordinary differential equations
Peer methods are multi-stage two-step methods for the solution of ordinary initial value problems where each stage posseses essentially the same accuracy and stability properties.
The class has been introduced by B.A. Schmitt and R. Weiner in SINUM 42 (2004).
In the literature parallel and non-parallel versions for stiff and non-stiff problems have been discussed.
An updated list of references on Peer methods with links can be found below.
Software EPPEER: explicit parallel peer methods
Explicit Parallel PEER methods allow easy parallelization with OpenMP on current multicore PCs without parallelization of the right-hand side of the ODE.
A Fortran95/OpenMP implementation is available in the package EPPEER here.
Main features of the code are:
- choice between 7 methods with orders 3..9 (order = number of stages)
- no parallelization of ODE call by user required
- methods may use 2..8 parallel cores through OpenMP, e.g. with the free GNU gfortran compiler
Note: one may expect good speed-up only for large systems with expensive ODE calls (examples below)
- automatic stepsize control with absolute and relative tolerances
- continous output with full order of convergence (optional)
The EPPEER package contains the following files:
ivpepp.f90 | mandatory | contains integrator subroutine EPPEER and supporting subroutines
| ivprkp.f90 | mandatory | contains Runge-Kutta methods for first time step (starting procedure)
| mbod4h.f90 | replace | example ODE multi-body-problem
| bruss2h.f90 | replace | example ODE 2D-Brusselator
| ivp_pmain.f90 | replace | main program, driver solving ODEs with set of tolerances, writing log files
| man_epp.pdf | | Documentation/manual
| mbod.plt | | Gnuplot file generates efficiency diagram from log files after running MBOD4h examples
| brus.plt | | Gnuplot file generates efficiency diagram from log files after running BRUSSh examples
| | | | | | | | |
Releases
- 2012/12/07: fixed small bug (missing absolute value in error estimate), may have caused failure for small systems
- 2012/08/27: first release
EPPEER parallel performance on Intel i7 Quadcore
EPPEER comes with 2 test problems BRUSS2h (2D-reaction-diffusion equation with Brusselator, small diffusion constant) and MBOD4h (celestial multi-body problem with 400 masses, evaluation expensive).
The performance of EPPEER with OpenMP and the GNU gfortan compiler is shown for both examples and all peer methods (named eppsxx, where s is number of stages).
The methods were run with tolerances atol=rtol=1D-2,...,1D-12.
- Parallel runtimes on Quadcore for all peer methods:
|
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problem MBOD4h
| Brusselator BRUSS2h
|
The diagram for MBOD4h shows efficiency increasing with order but slightly decreasing accuracy relative to tolerance.
For the mildly stiff Brusselator higher order pays off only for sharp tolerances.
- Good parallel performance is seen for the expensive multi-body problem MBOD4h only.
Parallel and sequential runtimes compared with Runge-Kutta method DOPRI5:
|
|
|
epp5f3 and dopri5
| epp7f4 and dopri5
| epp9f2 and dopri5
|
peek speed-up 3.1
| peek speed-up 3.3
| peek speed-up 4.0
|
List of references on peer methods
Schmitt,B.A, and Weiner,R (2004)
Parallel two-step W-methods with peer variables
SIAM J. Numer. Anal.42, 265-282
| Weiner,R; Schmitt,B.A; Podhaisky, H. (2004)
Parallel 'peer' two-step W-methods and their application to MOL systems
Appl. Numer. Math.48, 425-439
| Podhaisky,H; Weiner,R; Schmitt,B.A (2005)
Rosenbrock-type 'peer' two-step methods
Appl. Numer. Math.53, 409-420
| Schmitt,B.A; Weiner,R; Erdmann,K (2005)
Implicit parallel peer methods for stiff initial value problems
Appl. Numer. Math.53, 457-470
| Schmitt,B.A; Weiner,R; Podhaisky,H (2005)
Multi-implicit peer two-step W-methods for parallel time integration
BIT45, 197-217
| Podhaisky,H; Weiner,R; Schmitt,B.A (2006)
Linearly-implicit two-step methods and their implementation in Nordsieck-form
Appl. Numer. Math.56, 374-387
| Jebens,S; Weiner,R; Podhaisky,H; Schmitt,B.A (2008)
Explicit multi-step peer methods for special second order differential equations
Appl. Math. Comput.202, 803-813
| Weiner,R; Biermann,K; Schmitt,B.A; Podhaisky,H (2008)
Explicit two-step peer methods
Comput. Math. Appl.55, 609-619
| Gerisch,Alf; Lang,J; Podhaisky,H; Weiner,R (2009)
High-order linearly implicit two-step peer - finite element methods for time-dependent PDes
Appl. Numer. Math.59, 624-638 (*DOI:10.1016/j.apnum.2008.03.017)
| Gottermeier,B; Lang,J (2009)
Adaptive two-step peer methods for incompressible Navier-Stokes equations
Numer.Math+Adv.Appl.09., 387-395
| Kulikov,G.Yu; Weiner,R (2009)
Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation
J.Comput.Appl.Math.233, 2351-2364
| Schmitt,B.A; Weiner,R; Jebens,S. (2009)
Parameter optimization for explicit parallel peer two-step methods
Appl. Numer. Math.59, 769-782 (DOI:10.1016/j.apnum.2008.03.013)
| Weiner,R; Schmitt,B.A; Podhaisky,H; Jebens,S (2009)
Superconvergent explicit two-step peer methods
J.Comput.Appl.Math.223, 753-764 (doi:10.1016/j.cam.2008.02.014)
| Gottermeier,B; Lang,J. (2010)
Adaptive two-step peer methods for thermally coupled imcompressible flow
ECCOMAS CFD2010
| Kulikov,G.Yu; Weiner,R (2010)
Global error control in implicit parallel peer methods
Russ.J.Numer.Anal.Math.Model.25, 131-146
| Kulikov,G.Yu; Weiner,R (2010)
Variable-stepsize interpolating explicit parallel peer methods with inherent global error control
SIAM J. Sci. Comput.32,1695-1723
| Kulikov,G.Yu; Weiner,R (2010)
Global error control in implicit parallel peer methods
Russ.J.Numer.Anal.Math.Model.25, 131-146
| Schmitt,B.A; Weiner,R (2010)
Parallel start for explicit parallel two-step peer methods
Numer. Algorithms 53, 363-381 (DOI:10.1007/s11075-009-9267-2)
| Calvo,M.; Montijano,J.I; Randez,L; Daele,M.Van (2011)
On the derivation of explicit two-step peer methods
Appl. Numer. Math.61, 395-409
| Jebens,S; Knoth,O; Weiner,R (2011)
Partially implicit peer methods for the compressible Euler equations
J.Comput.Phys.230, 4955-4974
| Kulikov,G.Yu; Weiner,R (2011)
Global error estimation and control in linearly-implicit parallel two-steppeer W-methods
J.Comput.Appl.Math.236, 1226-1239
| Beck,Steffen; Weiner,R; Podhaisky,H; Schmitt,B.A (2012)
Implicit peer methods for large stiff ODE systems
J.Appl.Math.+Comp.38, 389-406 (DOI:10.1007/s12190-011-0485-0)
| Jebens,S; Knoth,O; Weiner,R (2012)
Explicit two-step peer methods for the compressible Euler equations
Monthly_Weather_Review_137, 2380-2392
Jebens,S; Knoth,O; Weiner,R. (2012)
Linearly-implicit peer methods for the compressible Euler equations
Appl. Numer. Math.62, 1380-1392
| Schmitt,B.A (2012)
On algebraic stability of general linear methods and peer methods
Appl. Numer. Math.62, 1544-1553 (DOI:10.1016/j.apnum.2012.06.005)
| Steinebach,Gerd; Weiner,R (2012)
Peer methods for the one-dimensional shallow water equations with CWENO space discretization
Appl. Numer. Math.62, 1567-1578
| Weiner,R; El-Azab,T (2012)
Exponential peer methods
Appl. Numer. Math.62, 1335-1348
| Weiner,R; Kulikov,G.Yu; Podhaisky,H (2012)
Variable-stepsize doubly quasi-consistent parallel explicit peer methods with global error control
Appl. Numer. Math.62, 1591-1603
| Schmitt,B.A; Kostina,E. (2012)
Peer two-step methods with embedded sensitivity approximation for parameter-dependent ODEs
SIAM J. Numer. Anal. 50, No. 5, 2182-2207.
| Schmitt,B.A; Weiner,R; Beck,S. (2013)
Two-step peer methods with continuous output
BIT 53, 717-739 (DOI:10.1007/s10543-012-0415-z)
| Schmitt,B.A. (2014)
Peer methods with improved embedded sensitivities for parameter-dependent ODEs
J.Comput.Appl.Math. 256, 242-253 (Reprint in JCAM 262, 25-36)
| Schröder,D.; Lang,J; Weiner, R. (2014)
Stability and consistency of discrete adjoint implicit peer methods
J.Comput.Appl.Math. 262, 73-86.
| Weiner,R; Kulikov, G; (2014)
Local and global error estimation and control within explicit two-step peer triples
J.Comput.Appl.Math. 262, 261-270.
| Beck,S.; Gonzalez-Pinto,S; Perez-Rodriguez,S.; Weiner, R. (2014)
A comparison of AMF- and Krylov methods in Matlab for large stiff ODE systems
J.Comput.Appl.Math. 262, 292-383.
| Weiner,R; Kulikov,G; (2014)
Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods
Copm.Math&Math.Physics 54, 604-619
| Weiner,R; Kulikov,G; (2014)
A singly diagonally implicit two-step peer triple with global error control for stiff ordinary differential equations
SIAM J.Sci.Comput. 37, A1593-A1613
| Schmitt,B.A. (2015)
Algebraic criteria for A-stability of peer two-step methods
arXiv:1506.05738
| Calvo,M; Montijano,J.I; Randez,L; Van Daele, M; (2015)
Exponentially fitted fifth-order two-step peer explicit methods
AIP Conference Proceedings 1648, 150015, doi:10.1063/1.4912445
| Horvath,Z; Podhaisky,H; Weiner,R; (2016)
Strong stability preserving explicit peer methods
J. Comput. Appl. Math. 296, 776-788,
doi:10.1016/j.cam.2015.11.005
| Weiner,R; Bruder,J; (2016)
Exponential Krylov peer integrators
BIT 56, 375-393,
doi:10.1007/s10543-015-0553-1
| Soleimani,B; Weiner,R; (2016)
A class of implicit peer methods for stiff systems
J. Comput. Appl. Math., doi:10.1016/j.cam.2016.06.014
| Schröder,D; Gerisch,A; Lang,J; (2016)
Space-time adaptive linearly implicit peer methods for parabolic problems
J. Comput. Appl. Math., doi:10.1016/j.cam.2016.08.023
| Schmitt,B.A; Weiner,R; (2016)
Efficient A-stable peer two-step methods
J. Comput. Appl. Math., doi:10.1016/j.cam.2016.08.045
| Schmitt,B.A; (2016)
Data sets for verification of A-stability and zero stability of peer two-step methods,
Mendeley Data, v1 http://dx.doi.org/10.17632/7393kvzn33.1
The data set contains Maxima worksheets and PDF output with additional data and the computations
for verification of the algebraic criteria for A- and zero stability of the 4 new peer methods from
the paper "Efficient A-stable peer two-step methods".
| Korch,M; Rauber,T; Stachowski, M; Werner, T; (2016)
Influence of locality on the scalability of method- and system-parallel explicit peer methods
Annals of Computer Science and Information Systems 8, 685-694, doi:10.15439/2016F464
| Lang, J.; Hundsdorfer, W; (2017)
Extrapolation-based implicit-explicit Peer methods with optimised stability regions
J. Comput. Phys. 337, 203-215,
doi:10.1016/j.jcp.2017.02.034
| Soleimani, B.; Knoth, O.; Weiner, R.; (2017)
IMEX peer methods for fast-wave - slow-wave problems.
Appl. Numer. Math. 118, 221 - 237,
doi:10.1016/j.apnum.2017.02.016
| Klinge, M.; Weiner, R.; Podhaisky, H.; (2017)
Optimally zero stable explicit peer methods with variable nodes.
BIT 58, 331-345, doi:10.1007/s10543-017-0691-8
| Weiner, R; Kulikov,G.Yu; Beck, S.; Bruder, J; (2017)
New third- and fourth-order singly diagonally implicit two-step peer triples with local and global error controls for solving stiff ordinary differential equations
J.Comput.Appl.Math. 316, 380-391.
doi:10.1016/j.cam.2016.06.013
| Klinge, M; Weiner, R.; (2018)
Strong stability preserving explicit peer methods for discontinuous Galerkin discretizations
J. Sci Comput 75, 1057-1078, doi:10.1007/s10915-017-0573-x
| Korch, M; Werner, T.; (2018)
Accelerating explicit ODE methods on GPUs by kernel fusion
Concurrency Computat Pract Exper. 2018;e4470. doi:10.1002/cpe.4470
| Kulikov, G.Y; Weiner, R. (2018)
Doubly quasi-consistent fixed-stepsize numerical integration of stiff ordinary differential equations with implicit two-step peer methods
J. Comput.Appl.Math. 340, 256-275,
doi:10.1016/j.cam.2018.02.037
| Klinge, M; Weiner, R.; Podhaisky, H; (2018)
Optimally zero stable explicit peer methods with variable nodes
BIT 58, 331-345, doi:10.1515/jnma-2017-0102
| Schneider, M; Lang, J.; Hundsdorfer, W.H; (2018)
Extrapolation-based super-convergent implicit-explicit Peer methods with A-stable implicit part
J. Comput. Phys. 367, 121-133,
doi:10.1016/j.jcp.2018.04.006
| Benner, P; Lang, N.; (2018)
Peer methods for the solution of large-scale differential matrix equations
arXiv:1807.08524
| Conte, D.; D'Ambrosio, R.D.; Moccaldi, M; Paternoster, B.(2018)
Adapted explicit two-step peer methods
J. Numerical Mathematics 27, doi:10.1515/jnma-2017-0102
| Massa, F.C; Noventa, G.; Lorini,M; Bassi,F; Ghidoni,A; (2018)
High-order linearly implicit two-step peer schemes for the discontinuous Galerkin solution of the incompressible Navier-Stokes equation
Computers&Fluids 162:55-71, doi:10.1016/j.compfluid.2017.12.003
| Montijano, J.I; Podhaisky, H.; Randez, L; Calvo, M. (2019)
A family of L-stable singly implicit Peer methods for solving stiff IVPs
BIT 59, 483–502,
doi:10.1007/s10543-018-0734-9
| Conte, D.; Moradi, D.; Paternoster, B.; Mohammedi, F; (2019)
Construction of exponentially fitted explicit peer methods
Int J Circuits Syst Signal Process 13:501–506~13:501-506
| Schneider, M; Lang, J.; Weiner, R; (2019)
Super-convergent implicit-explicit Peer methods with variable step sizes
J. Comput. Appl. Math,
doi:10.1016/j.cam.2019.112501
| Conte, D.; Mohammadi, F.; Moradi, L; Paternoster, B.(2020)
Exponentially fitted two-step peer methods for oscillatory problems
Comput. Appl. Math. 39, Nr.174,
doi:10.1007/s40314-020-01202-x
| Kulikov, G.Yu; Weiner, R.; (2020)
Variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs for solving stiff ordinary differential equations
Appl. Numer. Math. 154, 223-242, doi:10.1016/j.apnum.2020.04.003
| Schneider, M; Lang, J; Weiner, R.; (2021)
Super-convergent implicit-explicit Peer methods with variable step sizes
J. Comput. Appl. Math 387, 112501, doi:10.1016/j.cam.2019.112501
| Klinge, M; Hernandez-Abreu, D; Weiner, R.; (2021)
A comparison of one-step and two-step W-methods and peer methods with approximate matrix factorization
J. Comput. Appl. Math 387, 112519, doi:10.1016/j.cam.2019.112519
| Conte, D.; Pagano, G.; Paternoster, B.; (2021)
Jacobian-dependent two-stage peer method for ordinary differential equations
Computational Science and Its Applications ICCSA 2021, 309-324
| Lang, J.; Schmitt, B.A.; (2022)
Discrete adjoint implicit peer methods in optimal control
J.Comput.Appl.Math. 416, 114596,
doi:10.1016/j.cam.2022.114596
| Conte, D.; Pagano, G.; Paternoster, B.; (2022)
Two-step peer methods with equation-dependent coefficients
Comput. Appl. Math. 41, #140(2022),
doi:10.1007/s40314-022-01844-z
| Conte, D.; DeLuca, P.; Galietti, A; Giunta, G; Marcellino, L; Pagano, G; Paternoster, B.; (2022)
First experiences on parallelizing peer methods for numerical solution of a vegetation model
Computational Science and Its Applications ICCSA 2022, 384-394
| Lang, J.; Schmitt, B.A.; (2022)
Implicit A-stable peer triplets for ODE constrained optimal control problems
Algorithms 2022, 15, 310,
doi:10.3390/a15090310
| Lang, J.; Schmitt, B.A.; (2022)
A stiff MOL boundary control problem for the 1D heat equation with exact discrete solution
arXiv:2209.14051. Published as:
Lang, J.; Schmitt, B.A.; (2023)
Exact discrete solutions of boundary control problems for the 1D heat equation
J. Optim. Theory Appl., doi:10.1007/s10957-022-02154-4
| Abdi, A; Hojjati, G; Jackiewicz, Z; Pohaisky, H; Sharifi, M (2023)
On the implementation of explicit two-step peer methods with Runge–Kutta stability
Appl. Numer. Math,
doi:10.1016/j.apnum.2023.01.015
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