The high quality COLSYS/COLNEW collocation software package is widely used for the numerical solution of boundary value ODEs (BVODEs), often through interfaces to computing environments such as Scilab, R, and Python. The continuous collocation solution returned by the code is much more accurate at a set of mesh points that partition the problem domain than it is elsewhere; the mesh point values are said to be superconvergent. In order to improve the accuracy of the continuous solution approximation at non-mesh points, when the BVODE is expressed in first order system form, an approach based on continuous Runge-Kutta (CRK) methods has been used to obtain a superconvergent interpolant (SCI) across the problem domain. Based on this approach, recent work has seen the development of a new, more efficient version of COLSYS/COLNEW that returns an error controlled SCI.
However, most systems of BVODEs include higher derivatives and a feature of COLSYS/COLNEW is that it can directly treat such mixed order BVODE systems, resulting in improved efficiency, continuity of the approximate solution, and user convenience. In this paper we generalize the approach mentioned above for first order systems to obtain SCIs for collocation solutions of mixed order BVODE systems. The main contribution of this paper is the derivation of generalizations of continuous Runge-Kutta-Nyström methods that form the basis for SCIs for this more general problem class. We provide numerical results that (ⅰ) show that the SCIs are much more accurate than the collocation solutions at non-mesh points, (ⅱ) verify the order of accuracy of these SCIs, and (ⅲ) show that the cost of utilizing the SCIs is a small fraction of the cost of computing the collocation solution upon which they are based.
Citation: M. Adams, J. Finden, P. Phoncharon, P. H. Muir. Superconvergent interpolants for Gaussian collocation solutions of mixed order BVODE systems[J]. AIMS Mathematics, 2022, 7(4): 5634-5661. doi: 10.3934/math.2022312
The high quality COLSYS/COLNEW collocation software package is widely used for the numerical solution of boundary value ODEs (BVODEs), often through interfaces to computing environments such as Scilab, R, and Python. The continuous collocation solution returned by the code is much more accurate at a set of mesh points that partition the problem domain than it is elsewhere; the mesh point values are said to be superconvergent. In order to improve the accuracy of the continuous solution approximation at non-mesh points, when the BVODE is expressed in first order system form, an approach based on continuous Runge-Kutta (CRK) methods has been used to obtain a superconvergent interpolant (SCI) across the problem domain. Based on this approach, recent work has seen the development of a new, more efficient version of COLSYS/COLNEW that returns an error controlled SCI.
However, most systems of BVODEs include higher derivatives and a feature of COLSYS/COLNEW is that it can directly treat such mixed order BVODE systems, resulting in improved efficiency, continuity of the approximate solution, and user convenience. In this paper we generalize the approach mentioned above for first order systems to obtain SCIs for collocation solutions of mixed order BVODE systems. The main contribution of this paper is the derivation of generalizations of continuous Runge-Kutta-Nyström methods that form the basis for SCIs for this more general problem class. We provide numerical results that (ⅰ) show that the SCIs are much more accurate than the collocation solutions at non-mesh points, (ⅱ) verify the order of accuracy of these SCIs, and (ⅲ) show that the cost of utilizing the SCIs is a small fraction of the cost of computing the collocation solution upon which they are based.
| [1] |
M. Adams, C. Tannahill, P. H. Muir, Error control Gaussian collocation software for boundary value ODEs and 1D time-dependent PDEs, Numer. Algorithms, 81 (2019), 1505–1519. https://doi.org/10.1007/s11075-019-00738-2 doi: 10.1007/s11075-019-00738-2
|
| [2] |
U. M. Ascher, J. Christiansen, R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comp., 33 (1979), 659–679. https://doi.org/10.1090/S0025-5718-1979-0521281-7 doi: 10.1090/S0025-5718-1979-0521281-7
|
| [3] |
U. M. Ascher, J. Christiansen, R. D. Russell, Collocation software for boundary value ODEs, ACM Trans. Math. Softw., 7 (1981), 209–222. https://doi.org/10.1145/355945.355950 doi: 10.1145/355945.355950
|
| [4] | U. M. Ascher, R. M. M. Mattheij, R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Applied Mathematics Series, Philadelphia: Society for Industrial and Applied Mathematics, 1995. |
| [5] |
U. M. Ascher, R. J. Spiteri, Collocation software for boundary value differential-algebraic equations, SIAM J. Sci. Comp., 15 (1994), 938–952. https://doi.org/10.1137/0915056 doi: 10.1137/0915056
|
| [6] |
G. Bader, U. M. Ascher, A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Stat. Comp., 8 (1987), 483–500. https://doi.org/10.1137/0908047 doi: 10.1137/0908047
|
| [7] |
K. Burrage, F. H. Chipman, P. H. Muir, Order results for mono-implicit Runge-Kutta methods, SIAM J. Numer. Anal., 31 (1994), 876–891. https://doi.org/10.1137/0731047 doi: 10.1137/0731047
|
| [8] |
J. R. Cash, A. Singhal, Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential systems, IMA J. Numer. Anal., 2 (1982), 211–227. https://doi.org/10.1093/imanum/2.2.211 doi: 10.1093/imanum/2.2.211
|
| [9] |
J. F. L. Duval, E. Rotureau, Dynamics of metal uptake by charged soft biointerphases: impacts of depletion, internalisation, adsorption and excretion, Phys. Chem. Chem. Phys., 16 (2014), 7401–7416. https://doi.org/10.1039/C4CP00210E doi: 10.1039/C4CP00210E
|
| [10] |
W. H. Enright, K. R. Jackson, S. P. Nørsett, P. G. Thomsen, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Softw., 12 (1986), 193–218. https://doi.org/10.1145/7921.7923 doi: 10.1145/7921.7923
|
| [11] |
W. H. Enright, P. H. Muir, Efficient classes of Runge-Kutta methods for two-point boundary value problems, Computing, 37 (1986), 315–334. https://doi.org/10.1007/BF02251090 doi: 10.1007/BF02251090
|
| [12] | W. H. Enright, P. H. Muir, A Runge-Kutta Type Boundary Value ODE Solver with Defect Control, Technical Report 93-267, Department of Computer Science, University of Toronto, Toronto, 1993. |
| [13] |
W. H. Enright, P. H. Muir, Superconvergent interpolants for the collocation solution of boundary value ordinary differential equations, SIAM J. Sci. Comp., 21 (1999), 227–254. https://doi.org/10.1137/S1064827597329114 doi: 10.1137/S1064827597329114
|
| [14] |
W. H. Enright, R. Sivasothinathan, Superconvergent interpolants for collocation methods applied to mixed order BVODEs, ACM Trans. Math. Softw., 26 (2000), 323–351. https://doi.org/10.1145/358407.358410 doi: 10.1145/358407.358410
|
| [15] |
J. M. Fine, Interpolants for Runge-Kutta-Nyström methods, Computing, 39 (1987), 27–42. https://doi.org/10.1007/BF02307711 doi: 10.1007/BF02307711
|
| [16] | A. D. Garnadi, P. D. R. Lestari, Modeling hot water bath treatment of fruit using lateral method of lines in SCILAB, Preprint 2020060054, 2020. https://doi.org/10.20944/preprints202006.0054.v1 |
| [17] |
K. R. Green, R. J. Spiteri, Extended BACOLI: Solving one-dimensional multiscale parabolic PDE systems with error control, ACM Trans. Math. Softw., 45 (2019), 1–19. https://doi.org/10.1145/3301320 doi: 10.1145/3301320
|
| [18] | E. Hairer, S. P., Nörsett, G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, Second edition, Springer Series in Computational Mathematics, 8, Berlin: Springer-Verlag, 1993. |
| [19] |
N. J. Higham, The numerical stability of Barycentric Lagrange interpolation, IMA J. Numer. Anal., 24 (2004), 547–556. https://doi.org/10.1093/imanum/24.4.547 doi: 10.1093/imanum/24.4.547
|
| [20] |
H. Jin, S. Pruess, Uniformly superconvergent approximations for linear two-point boundary value problems, SIAM J. Numer. Anal., 35 (1998), 363–375. https://doi.org/10.1137/S0036142996297205 doi: 10.1137/S0036142996297205
|
| [21] |
S. Karlin, J. M. Karon, On Hermite-Birkhoff interpolation, J. Approx. Theory, 6 (1972), 90–115. https://doi.org/10.1016/0021-9045(72)90085-8 doi: 10.1016/0021-9045(72)90085-8
|
| [22] |
Z. Li, P. Muir, B-Spline Gaussian collocation software for two-dimensional parabolic PDEs, Adv. Appl. Math. Mech., 5 (2013), 528–547. https://doi.org/10.4208/aamm.13-13S09 doi: 10.4208/aamm.13-13S09
|
| [23] |
A. Marthinsen, Continuous extensions to Nyström methods for second order initial value problems, BIT, 36 (1996), 309–332. https://doi.org/10.1007/BF01731986 doi: 10.1007/BF01731986
|
| [24] |
A. Marunovic, M. Murkovic, A novel black hole mimicker: a boson star and a global monopole nonminimally coupled to gravity, Class. Quantum Grav., 31 (2014), 045010. https://doi.org/10.1088/0264-9381/31/4/045010 doi: 10.1088/0264-9381/31/4/045010
|
| [25] |
P. Muir, B. Owren, Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes, Math. Comp., 61 (1993), 675–699. https://doi.org/10.1090/S0025-5718-1993-1195425-8 doi: 10.1090/S0025-5718-1993-1195425-8
|
| [26] | P. H. Muir, M. Adams, Mono-implicit Runge-Kutta-Nyström methods with application to boundary value ordinary differential equations, BIT, 41 (2001), 776–799. |
| [27] | P. H. Muir, M. Adams, J. Finden, P. Phoncharon, Improving the Accuracy of Collocation Solutions of Mixed First and Second Order Boundary Value ODE Systems through the use of Superconvergent Interpolants, Technical Report 2019_002, Department of Mathematics and Computing Science, Saint Mary's University, 2019. |
| [28] |
B. Owren, M. Zennaro, Order barriers for continuous explicit Runge-Kutta methods, Math. Comp., 56 (1991), 645–661. https://doi.org/10.1090/S0025-5718-1991-1068811-2 doi: 10.1090/S0025-5718-1991-1068811-2
|
| [29] |
B. Owren, M. Zennaro, Derivation of optimal continuous explicit Runge-Kutta methods, SIAM J. Sci. Stat. Comp., 13 (1992), 1488–1501. https://doi.org/10.1137/0913084 doi: 10.1137/0913084
|
| [30] |
F. M. Pereira, S. C. Oliveira, Occurrence of dead core in catalytic particles containing immobilized enzymes: analysis for the Michaelis-Menten kinetics and assessment of numerical methods, Bioprocess Biosyst. Eng., 39 (2016), 1717–1727. https://doi.org/10.1007/s00449-016-1647-0 doi: 10.1007/s00449-016-1647-0
|
| [31] | N. Petit, A. Sciarretta, Optimal drive of electric vehicles using an inversion-based trajectory generation approach, Proceedings of the 18th World Congress, The International Federation of Automatic Control, Milano, Italy, (2011), 14519–14526. https: //doi.org/10.3182/20110828-6-IT-1002.01986 |
| [32] |
J. Pew, Z. Li, C. Tannahill, P. Muir, G. Fairweather, Performance analysis of error-control B-spline Gaussian collocation software for PDEs, Comput. Math. Appl., 77 (2019), 1888–1901. https://doi.org/10.1016/j.camwa.2018.11.025 doi: 10.1016/j.camwa.2018.11.025
|
| [33] |
S. Pruess, Interpolation schemes for collocation solutions of two point boundary value problems, SIAM J. Sci. Stat. Comp., 7 (1986), 322–333. https://doi.org/10.1137/0907021 doi: 10.1137/0907021
|
| [34] |
S. Pruess, H. Jin, A stable high-order interpolation scheme for superconvergent data, SIAM J. Sci. Comp., 17 (1996), 714–724. https://doi.org/10.1137/S1064827593257481 doi: 10.1137/S1064827593257481
|
| [35] |
M. Shakourifar, W. H. Enright, Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay, BIT, 52 (2012), 725–740. https://doi.org/10.1007/s10543-012-0373-5 doi: 10.1007/s10543-012-0373-5
|
| [36] |
J. H. Verner, Differentiable interpolants for high-order Runge-Kutta methods, SIAM J. Numer. Anal., 30 (1993), 1446–1466. https://doi.org/10.1137/0730075 doi: 10.1137/0730075
|
M. Adams, J. Finden, P. Phoncharon, P. H. Muir. Superconvergent interpolants for Gaussian collocation solutions of mixed order BVODE systems[J]. AIMS Mathematics, 2022, 7(4): 5634-5661. doi: 10.3934/math.2022312
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