Maximum norm stability of difference approximations to the mixed initial boundary-value problem for the heat equation
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- by J. M. Varah PDF
- Math. Comp. 24 (1970), 31-44 Request permission
Abstract:
We consider the heat equation ${u_t} = {u_{xx}}$ in the quarter-plane $x \geqq 0$, $t \geqq 0$ with initial condition $u(x,0) = f(x)$ and boundary condition $\alpha u(0,t) + {u_x}(0,t) = 0$. We are concerned with the stability of difference approximations ${\upsilon _\nu }^{n + 1} = Q{\upsilon _\nu }^n$ to this problem. Using the resolvent operator ${(Q - zI)^{ - 1}}$, we give sufficient conditions for consistent, onestep explicit schemes to be stable in the maximum norm.References
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
- Fritz John, Lectures on advanced numerical analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR 0221721
- Heinz-Otto Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 241010, DOI 10.1090/S0025-5718-1968-0241010-7
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- Olof B. Widlund, Stability of parabolic difference schemes in the maximum norm, Numer. Math. 8 (1966), 186–202. MR 196965, DOI 10.1007/BF02163187
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Math. Comp. 24 (1970), 31-44
- MSC: Primary 65.68
- DOI: https://doi.org/10.1090/S0025-5718-1970-0260215-1
- MathSciNet review: 0260215