Stellenbosch University
A provably stable and high order accurate solution of
the Frank-Kamenetskii partial differential
equation
C. Harley\(^*\), Data Science
Across Disciplines Research Group
(Institute for the Future of Knowledge), University of
Johannesburg,
E. Momoniat, Data Science across
Disciplines Research Group
(Institute for the Future of Knowledge), University of
Johannesburg
J. Nordström, Department of
Mathematics, Linköping University, Sweden
Department of Mathematics and Applied Mathematics, University of
Johannesburg
SAMS Subject Classification Number: 23, 20
The Frank-Kamenetskii partial differential equation (FKPDE) [1], \[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{k}{x} \frac{\partial u}{\partial x} + \exp(u), \qquad{x \in [0,1]}, \label{eq:1}\qquad{(1)}\] models a thermal explosion in a vessel. The shape factor, \(k\), describes the shape of the vessel where \(k=0\) for a slab, \(k=1\) for a cylinder and \(k=2\) for a sphere. The variable, \(u\), is the dimensionless temperature, \(x\) is the dimensionless spatial coordinate, and \(t\) is the dimensionless time. The boundary conditions are given by \[\left. \frac{\partial u}{\partial x} \right|_{x=0}=0, \qquad u(1,t)=0.\qquad{(2)}\] An investigation of asymptotic solutions at the boundary \(x=0\) proves that the solution \(u(x,t)\) is defined at the boundary in spite of the singularity that arises. We employ a multi-domain numerical procedure, where we utilise the Galerkin method on the domain \([0, \epsilon ]\) and the finite difference method on the domain \([\epsilon, 1]\). The SBP-SAT (summation by parts - simultaneous approximation term ) [2,3,4] methodology assists us in coupling these two numerical schemes at \(x=\epsilon\), so that we end up deriving provably stable and convergent numerical schemes for solving equation ([eq:1]). Results obtained in this paper can be applied to the Navier-Stokes equations in a cylindrical coordinate system.
References
[1] D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969.
[2] H.-O. Kreiss, G. Scherer, Finite element and finite difference methods for hyperbolic partial differential equations, in: C. De Boor (Ed.), Mathematical Aspects of Finite Elements in Partial Differential Equation, Academic Press, New York, 1974.
[3] B. Strand, Summation by parts for finite difference approximation for d/dx, J. Comput. Phys. 110 (1994) 47-–67.
[4] M. H. Carpenter, J. Nordstrom, D. Gottlieb, A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy, Journal of Computational Physics 148 (1999), 341–-365.
