A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method. (English) Zbl 1189.65222
The authors propose a numerical method to approximate solutions of diffusion transport equations in one dimensional space. The algorithm combines the B-spline Galerkin method, the Fourier transform and Gauss-Hermite quadrature formulation. The main results of the paper are concerned with the existence, uniqueness and upper error bounds of the solution to the discretized problem. Numerical experiments performed in the final part of the paper support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
diffusion transport equation; Galerkin method; Interpolation; Gauss-Hermite quadrature formulation; error bound; algorithm; B-spline; Fourier transform; numerical experimentsReferences:
| [1] | (Abramowitz, M.; Setgun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing, vol. 890 (1972), Dover: Dover New York) · Zbl 0543.33001 |
| [2] | Brattka, V.; Yoshikawa, A., Towards computability of elliptic boundary value problems in variational formulation, J. Complexity, 22, 858-880 (2006) · Zbl 1126.03052 |
| [3] | Brézis, H., Analyse Fonctionnelle: Théorie et Applications (1999), Dunod: Dunod Paris · Zbl 0511.46001 |
| [4] | de Boor, C., A Practical Guide to Splines (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0406.41003 |
| [5] | Klose, A. D.; Larsen, E. W., Light transport in biological tissue based on the simplified spherical harmonics equations, J. Comput. Phys., 220, 441-470 (2006) · Zbl 1122.78015 |
| [6] | Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications, vol. I (1972), Springer: Springer New York · Zbl 0223.35039 |
| [7] | Natterer, F.; Sielschott, H.; Dorn, O.; Dierkes, T.; Palamodov, V., Fréchet derivatives for some bilinear inverse problems, SIAM J. Appl. Math., 62 6, 2092-2113 (2002) · Zbl 1010.35115 |
| [8] | Novikov, R. G., \( \overline{\partial} \)-method with nonzero background potential application to inverse scattering for the two-dimensional acoustic equation, Commun. Partial Diff. Eqns., 21, 34, 597-618 (1996) · Zbl 0855.35130 |
| [9] | Schultz, M. H.; Varga, R. S., L-splines, Numer. Math., 10, 345-369 (1967) · Zbl 0183.44402 |
| [10] | Schultz, M. H., Splines Analysis (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0333.41009 |
| [11] | Seaïd, M., Multigrid Newton Krylov method for radiation in diffusive semitransparent media, J. Comput. Appl. Math., 203, 498-515 (2007) · Zbl 1115.65128 |
| [12] | Teleaga, I.; Seaïd, M.; Gasser, I.; Klar, A.; Strukmeier, J., Radiation models for thermal flows at law Mach number, J. Comput. Phys., 205, 506-525 (2006) · Zbl 1099.76056 |
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