Numerical approximation of the quenching time for a non-Newtonian filtration equation with singular boundary flux
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This paper concerns the study of the numerical approximation for the following initial-boundary value problem .$$\left\{% \begin{array}{ll} \hbox{$u_t=\left( \vert u_{x}\vert^{p-2}u_{x}\right)_{x} + \left( 1-u\right)^{-h} ,\quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)=0, \quad u_{x}(1,t)= - u^{-q}(1,t),\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x) > 0,\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ where $p\geq 2$, $h>0$, $q>0$. $u_{0}: [0,1] \rightarrow (0,1)$ and satisfies compatiblity conditions. We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.
Abstract
This paper concerns the study of the numerical approximation for the following initial-boundary value problem .$$\left\{% \begin{array}{ll} \hbox{$u_t=\left( \vert u_{x}\vert^{p-2}u_{x}\right)_{x} + \left( 1-u\right)^{-h} ,\quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)=0, \quad u_{x}(1,t)= - u^{-q}(1,t),\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x) > 0,\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ where $p\geq 2$, $h>0$, $q>0$. $u_{0}: [0,1] \rightarrow (0,1)$ and satisfies compatiblity conditions. We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.
참고문헌 (8)
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