Coupled sound and heat flow and the method of least squares

Author:
Alfred Carasso

Journal:
Math. Comp. **29** (1975), 447-463

MSC:
Primary 65M15

MathSciNet review:
0395252

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Abstract: We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain in , with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-squares method is devised, using trial functions which need not satisfy the homogeneous boundary conditions. Certain unknown normal derivatives of the solution enter the boundary integrals. By using suitable weights, these unknown derivatives can be set equal to zero without impairing the accuracy of the Crank-Nicolson scheme. However, one must use smoother trial functions to obtain this accuracy.

**[1]**William F. Ames,*Numerical methods for partial differential equations*, Barnes & Noble, Inc., New York, 1969. MR**0263257****[2]**G. Birkhoff, M. H. Schultz, and R. S. Varga,*Piecewise Hermite interpolation in one and two variables with applications to partial differential equations*, Numer. Math.**11**(1968), 232–256. MR**0226817****[3]**J. H. Bramble and S. R. Hilbert,*Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation*, SIAM J. Numer. Anal.**7**(1970), 112–124. MR**0263214****[4]**James H. Bramble and Vidar Thomée,*Semidiscrete least-squares methods for a parabolic boundary value problem*, Math. Comp.**26**(1972), 633–648. MR**0349038**, 10.1090/S0025-5718-1972-0349038-4**[5]**Alfred Carasso,*A least squares procedure for the wave equation*, Math. Comp.**28**(1974), 757–767. MR**0373310**, 10.1090/S0025-5718-1974-0373310-7**[6]**F. HARLOW & A. AMSDEN,*Fluid Dynamics*, LASL Monograph LA 4700, Los Alamos Scientific Laboratories, Los Alamos, N. M., 1971.**[7]**J- L. Lions,*Sur l’approximation de la solution d’équations d’évolution couplées*, Rend. Mat. (6)**1**(1968), 141–176 (French, with Italian summary). MR**0239298****[8]**J.-L. Lions,*On the numerical approximation of some equations arising in hydrodynamics*, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 11–23. MR**0261180****[9]**J.-L. Lions and P. A. Raviart,*Remarques sur la résolution et l’approximation d’équations d’évolution couplées*, ICC Bull.**5**(1966), 1–21 (French). MR**0204812****[10]**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243****[11]**Hirofumi Morimoto,*Stability in the wave equation coupled with heat flow*, Numer. Math.**4**(1962), 136–145. MR**0154437****[12]**R. D. RICHTMYER & K. W. MORTON,*Stability Studies for Difference equations*. I.*Non-Linear Instability*. II.*Coupled Sound and Heat Flow*, Report NYO 1480-5, Courant Inst. of Math. Sci., New York Univ., New York, 1964.**[13]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[14]**Carl A. Rouse,*A method for the numerical calculation of hydrodynamic flow and radiation diffusion by implicit differencing*, J. Soc. Indust. Appl. Math.**9**(1961), 127–135. MR**0129557****[15]**S. M. SERBIN, Doctorial Dissertation, Cornell Univ., Ithaca, N. Y., 1972.**[16]**R. S. Varga,*Functional analysis and approximation theory in numerical analysis*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3. MR**0310504****[17]**B. WENDROFF,*The Initial Value Problem*, Lecture Notes, Department of Mathematics, University of Denver, Col., 1969.

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DOI:
https://doi.org/10.1090/S0025-5718-1975-0395252-4

Article copyright:
© Copyright 1975
American Mathematical Society