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.

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:

The EPPEER package contains the following files:
  • ivpepp.f90mandatorycontains integrator subroutine EPPEER and supporting subroutines
  • ivprkp.f90mandatorycontains Runge-Kutta methods for first time step (starting procedure)
  • mbod4h.f90replaceexample ODE multi-body-problem
  • bruss2h.f90replaceexample ODE 2D-Brusselator
  • ivp_pmain.f90replacemain program, driver solving ODEs with set of tolerances, writing log files
  • man_epp.pdfDocumentation/manual
  • mbod.pltGnuplot file generates efficiency diagram from log files after running MBOD4h examples
  • brus.pltGnuplot file generates efficiency diagram from log files after running BRUSSh examples


    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.

    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/
  • Gottermeier,B; Lang,J. (2010)
    Adaptive two-step peer methods for thermally coupled imcompressible flow
  • 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
  • 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/
  • 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/
  • Schröder,D; Gerisch,A; Lang,J; (2016)
    Space-time adaptive linearly implicit peer methods for parabolic problems
    J. Comput. Appl. Math., doi:10.1016/
  • Schmitt,B.A; Weiner,R; (2016)
    Efficient A-stable peer two-step methods
    J. Comput. Appl. Math., doi:10.1016/
  • Schmitt,B.A; (2016)
    Data sets for verification of A-stability and zero stability of peer two-step methods,
    Mendeley Data, v1
    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/
  • 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/
  • 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/
  • 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/
  • Benner, P; Lang, N.; (2018)
    Peer methods for the solution of large-scale differential matrix equations
  • 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/
  • 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
  • Lang, J.; Schmitt, B.A.; (2020)
    Discrete adjoint implicit peer methods in optimal control