An implicit fourth order difference method for viscous flows

Authors:
Daniel S. Watanabe and J. Richard Flood

Journal:
Math. Comp. **28** (1974), 27-32

MSC:
Primary 65N05

MathSciNet review:
0341892

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Abstract: A new implicit finite-difference scheme for viscous flows is presented. The scheme is based on Simpson's rule and two-point Hermite interpolation, is uniformly accurate to fourth order in time and space, and is unconditionally stable according to a Fourier stability analysis. Numerical solutions of Burger's equation are presented to illustrate the order and accuracy of the scheme.

**[1]**C. G. Broyden,*A new method of solving nonlinear simultaneous equations*, Comput. J.**12**(1969/1970), 94–99. MR**0245197****[2]**Samuel Z. Burstein and Arthur A. Mirin,*Third order difference methods for hyperbolic equations*, J. Computational Phys.**5**(1970), 547–571. MR**0282545****[3]**R. W. MacCormack, "Numerical solution of the interaction of a shock wave with a laminar boundary layer,"*Lecture Notes in Physics*, v. 8, Springer-Verlag, Berlin, 1971, pp. 151-163.**[4]**E. L. Rubin & S. Z. Burstein, "Difference methods for the inviscid and viscous equations of a compressible gas,"*J. Computational Phys.*, v. 2, 1967, pp. 178-196.**[5]**V. V. Rusanov,*On difference schemes of third order accuracy for non-linear hyperbolic systems*, J. Computational Phys.**5**(1970), 507–516. MR**0275699****[6]**Gideon Zwas and Saul Abarbanel,*Third and fourth order accurate schemes for hyperbolic equations of conservation law form*, Math. Comp.**25**(1971), 229–236. MR**0303766**, 10.1090/S0025-5718-1971-0303766-4

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DOI:
https://doi.org/10.1090/S0025-5718-1974-0341892-7

Keywords:
Partial differential equations,
initial-value problems,
finite-difference schemes,
unconditional stability,
high order accuracy,
viscous flows

Article copyright:
© Copyright 1974
American Mathematical Society