Consider one initial boundary value problem for sphere.$$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f$$Here is explicit numerical scheme (we consider that it is stable):$$\frac{u^{n+1}_{i,j,k}-u^{n}_{i,j,k}}{\Delta t}=A_{i,j,k}{\frac{\partial u}{\partial r}}_{i,j,k,+/-}^{n}+B_{i,j,k}{\frac{\partial u}{\partial \varphi}}_{i,j,k,+/-}^{n}+C_{i,j,k}{\frac{\partial u}{\partial \psi}}_{i,j,k,+/-}^{n}+R_{i,j,k}{\frac{\partial^{2} u}{\partial r^2}}_{i,j,k}^{n}+\Phi_{i,j,k}{\frac{\partial^{2} u}{\partial \varphi^2}}_{i,j,k}^{n}+\Psi_{i,j,k}{\frac{\partial^{2} u}{\partial \psi^2}}_{i,j,k}^{n}+f^{n}_{i,j,k} ;$$$u(r,\varphi,\psi,t=0)=u_{0}(r,\varphi,\psi)$;$u(r=r_{0},\varphi,\psi,t)=0;$Under $+/-$ for operator it is understood that for operator was used upwind scheme; the operators are standart.It is clear that such scheme works for all domain, except points on $r=r_{0}$(actually, we don't need to compute values here, because they are known because of boundary condition) and for point $r=0$ in the center of the ball.Consider the following method for evaluating $u^{n+1}$ in the center of the ball: we use the definition of divergence$$\operatorname{div}(A(x,y,z))=\lim\limits_{R\to 0}\frac {\oint_{B_{R}(x,y,z)}{(A,n)ds}}{V_{R}}, $$therefore,we can write$$\frac{u^{n+1}_{1,j,k}-u^{n}_{1,j,k}}{\Delta t}=\frac {\oint_{B_{r_2}}{(A\nabla u,\overline{n})ds}}{V_{r_2}} +\overline{f}^{n}_{1,j,k} ,$$where $B_{r_2}$ -- a ball of radius equal to the value of unit step for $r$, $\overline{f}$ -- mean value of $f$ on the surface of $B_{r_2}$.Question: will this method be stable for the center of the point?If not, what stable method of computing the value in the center of ball you can offer?
Stability question (finite difference): dealing with corner nodes
pde;finite difference;stability