Complutense University Library

Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model

Díaz Díaz, Jesús Ildefonso and Bermejo, R. and Carpio, Jaime and Tello, J. Ignacio (2009) Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model. Mathematical and Computer Modelling, 49 (5-6). pp. 1180-1210. ISSN 0895-7177

[img] PDF
Restricted to Repository staff only until 31 December 2020.

2MB

Official URL: http://www.sciencedirect.com/science/article/pii/S0895717708001386

View download statistics for this eprint

==>>> Export to other formats

Abstract

The purpose of this paper is to carry out the mathematical and numerical analysis of a two-dimensional nonlinear parabolic problem on a compact Riemannian manifold without boundary, which arises in the energy balance for the averaged surface temperature. We use a possibly quasi-linear diffusion operator suggested by P. H. Stone in 1972. The modelling of the Budyko discontinuous coalbedo is formulated in terms of a bounded maximal monotone graph of R(2). The existence of global solutions is proved by applying a fixed point argument. Since the uniqueness of solutions may fail for the case of discontinuous coalbedo, we introduce the notion of non-degenerate solutions and show that the problem has at most one solution in this class of functions. The numerical analysis is carried out for the special case of a spherical Earth and uses quasi-uniform spherical triangles as finite elements. We study the existence, uniqueness and stability of the approximate solutions. We also show results of some long-term numerical experiments.


Item Type:Article
Uncontrolled Keywords:parabolic p-laplacian; approximation; equations; climate; nonlinear diffusive energy balance model; non-degenerate solution; finite elements; 2-sphere
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:15131
References:

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.

J.W. Barrett, W.B. Liu, Finite element approximation of the parabolic p-Laplacian, SIAM Journal on Numerical Analysis 31 (1994) 413_428.

J.R. Baumgardner, P.O. Frederickson, Icosahedral discretization of the two-sphere, SIAM Journal on Numerical Analysis 22 (1985) 1107_1115.

N. Boal, V. Domnguez, F.-J. Sayas, Asymptotic properties of some triangulations of the sphere, Journal of Computational and Applied Mathematics 211 (2008) 11_22.

S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Heidelberg, 2002.

H. Brezis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973.

M.I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus 21 (1969) 611_619.

P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.

I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984.

S.S. Chow, Finite element error estimates for the nonlinear elliptic equation of monotone type, Numerische Mathematik 54 (1989) 373_393.

J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, in: Pitman (Ed.), Elliptic Equations, vol. I, Boston, 1985.

J.I. Díaz, Mathematical analysis of some diffusive energy balance climate models, in the book, in: J.I. Díaz, J.L. Lions (Eds.), Mathematics, Climate and Environment, Masson, Paris, 1993, pp. 28_56.

J.I. Díaz (Ed.), The Mathematics of Models in Climatology and Environment, ASI NATO Global Change Series I, no. 48, Springer-Verlag, Heidelgerg, 1996.

J.I. Díaz, J. Hernández, L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology,

Journal of Mathematical Analysis and Applications 216 (1997) 593_613.

J.I. Díaz, G. Hetzer, A quasilinear functional reaction-diffusion equation arising in Climatology, in: Equations aux derivees partielles et applications:

Articles dedies a Jacques Louis Lions, Gautier Villards, Paris, 1998, pp. 461_480.

J.I. Díaz, G. Hetzer, L. Tello, An energy balance climate model with hysteresis, Nonlinear Analysis 64 (2006) 2053_2074.

G. Dziuk, Finite element for the Beltrami operator on arbitrary surfaces, in: Partial Differential Equations and Calculus of Variations, in: Lectures Notes in Mathematics, vol. 1357, Springer, Heidelberg, 1988, pp. 142_155.

J.I. Díaz, L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica 50 (1) (1999) 19_51.

M. Ghil, S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Springer-Verlag,

New York, 1987.

G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of

Mathematics 16 (1990) 203_216.

G. Hetzer, H. Jarausch, W. Mackens, A multiparameter sensitivity analysis of a 2D diffusive climate model, Impact and Computing in Science and

Engineering 1 (1989) 327_339.

N. Ju, Numerical Analysis of parabolic p-laplacian. Approximation of trajectories, SIAM Journal on Numerical Analysis 37 (2000) 1861_1884.

G. Lippold, Error estimate for the implicit Euler approximation of an evolution inequality, Nonlinear Analysis 15 (1990) 1077_1089.

G.R. North, J.A. Coakley, Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity, Journal of the Atmospheric Sciences 36 (1979) 1189_1203.

J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems, SIAM Journal on Numerical Analysis 33 (1996) 68_87.

G. Savare, Weak solutions and maximal regularity for abstract evolution inequalities, Advances in Math. Sci. and Appl. 6 (1996) 337_418.

W.D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, Journal of Applied Meteorology 8 (1969) 392_400.

P.H. Stone, A simplified radiative _ dynamical model for the static stability of rotating atmospheres, Journal of the Atmospheric Sciences 29 (1972)

405_418.

I.I. Vrabie, Compactness Methods for Nonlinear Equations, Pitman Longman, London, 1987.

D. Wei, Existence, Uniqueness and Numerical Analysis of solutions of a quasilinear parabolic problem, SIAM Journal on Numerical Analysis 29 (1992) 484_497.

X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Applicable Analysis 42 (1991) 33-59.

Deposited On:08 May 2012 08:53
Last Modified:06 Feb 2014 10:17

Repository Staff Only: item control page