We will be concerned with the numerical solution of the parabolic partial differential equation (math)and also the first order hyperbolic partial differential equation
$$\frac{{\partial u}}{{\partial t}} = L\left( {u,r,t,{D^2}} \right)u$$
where L is a linear operator (not the same operator in both equations),
$$u = {\left( {{u_1},{u_2},...{u_n}} \right)^T}D = \left( {\frac{\partial }{{\partial {x_1}}},\frac{\partial }{{\partial {x_2}}},...,\frac{\partial }{{\partial {x_S}}}} \right)r = {\left( {{x_1},{x_2},...,{x_S}} \right)^T}$$
and where the matrix coefficients in L may depend on u, r and t. We assume the problems are well posed.


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  1. 1.
    Courant, R., K. Friedrichs and H. Lewy: On the Partial Différence Equations of Mathematical Physics. IBM Journ. of research and development, 11 (1967), 213–234.CrossRefGoogle Scholar
  2. 2.
    D’jakonov, Ye. G.: On the application of disintegrating difference operators. Mat. i Mat.Fiz., 3 (1963), 385–388.Google Scholar
  3. 3.
    Fairweather, G. and A. R. Mitchell: A New Computational procedure for ADI methods. SIAM J. Num. Anal. 4 (1967), 163–170.CrossRefGoogle Scholar
  4. 4.
    Gourlay, A. R. and A. R. Mitchell: On the Structure of Alternating Direction Implizit (ADI) and Locally One Dimensional (LOD) methods. JIMA 9 (1972), 80–90.Google Scholar
  5. 5.
    Gourlay, A. R. and J. Ll. Morris: A Multistep Formulation of the Optimized Lax-Wendroff method for Nonlinear Hyperbolic Systems in two space variables. Maths. of Comp. 22(1968), 715–720.CrossRefGoogle Scholar
  6. 6.
    Gourlay, A. R. and J. Ll. Morris: On the Comparison of Multistep Formulations of the Optimized Lax-Wendroff Method for Nonlinear Hyperbolic Systems in Two Space Variables. J. Com. Phys. 5 (1970), 229–243.CrossRefGoogle Scholar
  7. 7.
    Gourlay, A. R. and J. Ll. Morris: Deferred approach to the limit in non-linear hyperbolic systems. Computer Journ. 11 (1968), 95–101.CrossRefGoogle Scholar
  8. 8.
    Gourlay, A. R., G. R. McGuire and J. Ll. Morris: One dimensional methods for the numerical solution of nonlinear hyperbolic systems. Conference on Applications of Numerical Analysis, Vol. 228 Lecture Notes in Mathematics, Springer-Verlag (1971).Google Scholar
  9. 9.
    Lawson, J. D. and J. Ll. Morris: Alternating Direction locally Dimensional Methods for partial Differential Equations in space variables. (1972).Google Scholar
  10. 10.
    McGuire, G. R. and J. Ll. Morris: Boundary techniques for the multistep formulation of the optimized Lax Wendroff method for nonlinear hyperbolic systems in two space variables, (to appear) JIMA, 197 2.Google Scholar
  11. 11.
    Mitchell, A. R.: Computational Methods in Partial Differential Equations. John Wiley and Sons (1969).Google Scholar
  12. 12.
    Morris, J. Ll.: On the Numerical Solution of a heat Equation associated with a thermal print head. J. Comp. Phys. 5 (1970), 208–228.CrossRefGoogle Scholar
  13. 13.
    Richtmyer, R.: A survey of difference methods for nonsteady fluid dynamics. NCAR Tech., Notes 63–2, (1963).Google Scholar
  14. 14.
    Samarskii, A. A.: An economic method of solving the multidimensional parabolic equation in an arbitrary region. Zh. Vych. Mat. Mat. Fiz. 2 (1962), 787–811.Google Scholar
  15. 15.
    Marshall-Smith, J.: M. Sc. Thesis, University of Dundee (1973).Google Scholar
  16. 16.
    Strang, W. G.: Accurate Partial Difference Methods II: Non linear problems. Num. Math. 6 (1964), 37–46.CrossRefGoogle Scholar
  17. 17.
    Strang, W. G.: On the construction and comparison of Difference Schemes. SIAM J. Numer. Anal. 5 (1968), 506–517.CrossRefGoogle Scholar
  18. 18.
    Wilson, J. C.: Stability of Richtmyer type difference schemes in any finite number of space variables and their comparison with multistep Strang schemes, (to appear) JIMA (1972).Google Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • J. Ll. Morris
    • 1
  1. 1.DundeeUK

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