The probability approach to numerical solution of nonlinear parabolic equations

2002 ◽  
Vol 18 (4) ◽  
pp. 490-522 ◽  
Author(s):  
G. N. Milstein
Keyword(s):  
Numerical Solution ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Probability Approach ◽  
Nonlinear Parabolic

scholarly journals Block monotone iterations for solving coupled systems of nonlinear parabolic equations

2020 ◽  
Vol 61 ◽  
pp. C166-C180
Author(s):  
Mohamed Saleh Mehdi Al-Sultani ◽  
Igor Boglaev
Keyword(s):  
Numerical Solution ◽  
Iterative Methods ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Coupled Systems ◽  
Parabolic Problems ◽  
Nonlinear Elliptic ◽  
Monotone Iterations ◽  
Nonlinear Parabolic ◽  
Appl Math

The article deals with numerical methods for solving a coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing. We employ block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of a solution are discussed. Numerical experiments are presented. References Al-Sultani, M. and Boglaev, I. ''Numerical solution of nonlinear elliptic systems by block monotone iterations''. ANZIAM J. 60:C79–C94, 2019. doi:10.21914/anziamj.v60i0.13986 Al-Sultani, M. ''Numerical solution of nonlinear parabolic systems by block monotone iterations''. Tech. Report, 2019. https://arxiv.org/abs/1905.03599 Boglaev, I. ''Inexact block monotone methods for solving nonlinear elliptic problems'' J. Comput. Appl. Math. 269:109–117, 2014. doi:10.1016/j.cam.2014.03.029 Lui, S. H. ''On monotone iteration and Schwarz methods for nonlinear parabolic PDEs''. J. Comput. Appl. Math. 161:449–468, 2003. doi:doi.org/10.1016/j.cam.2003.06.001 Pao, C. V. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992. doi:10.1007/s002110050168 Pao C. V. ''Numerical analysis of coupled systems of nonlinear parabolic equations''. SIAM J. Numer. Anal. 36:393–416, 1999. doi:10.1137/S0036142996313166 Varga, R. S. Matrix iterative analysis. Springer, Berlin, 2000. 10.1007/978-3-642-05156-2 Zhao, Y. Numerical solutions of nonlinear parabolic problems using combined-block iterative methods. Masters Thesis, University of North Carolina, 2003. http://dl.uncw.edu/Etd/2003/zhaoy/yaxizhao.pdf

Kinetics of oxidation of ammonia on cobalt catalyst I model of the reactor with catalytically active wall

1982 ◽  
Vol 47 (8) ◽  
pp. 2087-2096 ◽  
Author(s):  
Bohumil Bernauer ◽  
Antonín Šimeček ◽  
Jan Vosolsobě
Keyword(s):  
Parabolic Equations ◽  
Cobalt Catalyst ◽  
Nonlinear Parabolic Equations ◽  
Catalytic Reactions ◽  
Temperature Profiles ◽  
Dimensional Model ◽  
Kinetics Of Oxidation ◽  
Nonlinear Parabolic ◽  
Catalytically Active ◽  
Kinetics Of

A two dimensional model of a tabular reactor with the catalytically active wall has been proposed in which several exothermic catalytic reactions take place. The derived dimensionless equations enable evaluation of concentration and temperature profiles on the surface of the active component. The resulting nonlinear parabolic equations have been solved by the method of orthogonal collocations.

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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