Publication Cover
Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 41, 2002 - Issue 3-4
26
Views
13
CrossRef citations to date
0
Altmetric
Original Articles

THE TIME DIMENSION AND A UNIFIED MATHEMATICAL FRAMEWORK FOR FIRST-ORDER PARABOLIC SYSTEMS

Pages 239-262 | Published online: 30 Nov 2010

References

  • Crank , J. and Nicolson , P. 1947 . A Practical Method for the Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type . Proc. Cambridge Phil. Soc. , 43 : 50
  • Lees , H. 1966 . A Linear Three-Level Differences Scheme for Quasilinear Parabolic Equations . Mech. Comput. , 20 : 516
  • W. Liniger, Global Accuracy Stability of One and Two Step Integration Formulae for Stiff Ordinary Differential Equation, Conf. Numerical Solution of Differential Equations, Dundee University, Scotland, U.K. 1969.
  • Zienkiewicz , O. C. and Taylor , R. L. 1994 . The Finite Element Method , Vol. 1 , New York : McGraw-Hill .
  • Tamma , K. K. , Zhou , X. and Valasutean , R. 1997 . Computational Algorithms for Transient Analysis: The Burden of Weight and Consequences Towards Formalizing Discrete Numerically Assigned [DNA] Algorithmic Markers: Wp-Family . Comput. Meth. Appl. Mech. Eng. , 149 : 153
  • Tamma , K. K. , Zhou , X. and Sha , D. 1999 . Towards a Formal Theory of Development/ Evolution and Characterization of Time Discretized Operators for Heat Transfer . Int. J. Numer. Meth. Heat Fluid Flow , 9 : 348
  • Tamma , K. K. , Zhou , X. and Sha , D. 2000 . The Time Dimension: A Theory of Development/ Evolution, Classification, Characterization and Design of Computational Algorithms for Transient/Dynamic Applications . Arch. Comput. Mech. , 7 ( 2 ) : 67 – 290 .
  • Nersett , Syvert P. 1974 . One-Step Methods of Hermite Type for Numerical Integration of Stiff Systems . BIT , 14 : 63
  • Trujillo , D. M. 1975 . The Direct Numerical Integration of Linear Matrix Differential Equations Using Padé Approximations . Int. J. Numer. Meth. Eng. , 9 : 259
  • Zlamal , M. 1977 . Finite Elements Methods in Heat Conduction Problems , New York : Academic Press .
  • Argyris , J. H. , Vaz , L. E. and Willam , K.J. 1977 . Higher Order Methods for Transient Diffusion Analysis . Comput. Meth. Appl. Mech. Eng. , 12 : 243
  • Tamma , K. K. , Zhou , X. and Sha , D. 1999 . Transient Algorithms for Heat Transfer: General Developments and Approaches for Generating Nth-Order Time Accurate Operators Including Practically Useful Second-Order Forms . Int. J. Numer. Meth. Eng. , 44 : 1545
  • Delfour , M. , Hager , W. and Trochu , F. 1981 . Discontinous Galerkin Methods for Ordinary Differential Equations . Math. Comput. , 36 ( 154 ) April : 455 – 473 .
  • Bottasso , C. L. 1997 . A New Look at Finite Elements in Time: A Variational Interpretation of Runge-Kuta Methods . Appl. Numer. Math. , 25 : 355 – 368 .
  • R. Kanapady and K. K. Tamma, Unified High-Order Hierarchical State Space Time Discontinuous Operators for Computational Dynamics, Computer Modeling for Engineering and Sciences (in review).
  • Dahlquist , G. 1963 . A Special Stability Problem for Unear Multistep Methods . BIT , 3 : 27
  • X. Zhou, and K. K. Tamma, A Unified Theory Underlying Computational Algorithms for Linear First-Order Systems, Int. J. Numer. Meth. Eng., (in review).
  • Tarnow , N. and Simo , J. C. 1994 . How to Render Second-Order Accurate Time Stepping Algorithms Fourth-Order Accurate While Remaining the Stability and Conservation Properties . Comput. Meth. Appl. Mech. Eng. , 115 : 233
  • Fung , T. C. 1999 . Complex-Time-Step Methods for Transient Analysis . Int. J. Numer. Meth. Eng. , 46 : 1253
  • Donea , J. , Roig , B. and Huerta , A. 2000 . High-Order Accurate Time-Stepping Schemes for Convection-Diffusion Problems . Comput. Meth. Appl. Mech. Eng. , 182 : 249 – 275 .
  • B. L. Hulme, One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems, Math. Comput., vol. 26, no. 118, pp. 415-426, April 1972.
  • Butcher , J. C. 1964 . Implicit Runge-Kuta Processes . Math. Comput. , 18 : 50 – 64 .
  • Butcher , J. C. 1964 . Integration Processes Based on Radau Quadrature Formulas . Math. Campul. , 26 : 233 – 244 .
  • B. L. EhIe, On Fade Approximations to the Exponential Function and A-Stable Methods for the Numerical Solution of Initial Value Problems, Tech. Rep. CSRR 2010, Dept. AACS, University of Waterloo, Waterloo, Ontario, Canada, 1969.
  • Davis , P. J. 1975 . Interpolation and Approximation , New York : Dover .
  • Chipman , F. H. 1971 . A-stable Runge-Kutfc Processes . BIT , 11 : 384 – 388 .
  • R. Kanapady and K. K. Tamma, On a Novel Design and Unified Approach for Time Discontinuous/Continuous Operators and Equivalence, Finite Elements in Analysis and Design (in review).
  • Dupont , T. , Fairweather , G. and Johnson , P. 1974 . Three-Level Galerkin Methods for Parabolic Equations . SIAM J. , 11 : 392
  • Euler , L. 1913 . De Integratione Aequationum Differentialium per Appriximationem . Opera Omnia , 11 : 424
  • W. Liniger, Optimization of a Numerical Integration Method for Stiff Systems of Ordinary Differential Equations, IBM Res. Rep. RC2198, 1968.
  • Wood , W. L. 1990 . Practical Time Stepping Schemes , Oxford, U.K. : Clarendon Press .
  • C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations, in A. J. H. Morrell (ed.), Information Processing, vol. 68, p. 187, North Holland, Dordrecht, 1969.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.