A numerical procedure for diffusion subject to the specification of mass
The authors look for the numerical solution of the diffusion problem \(u_ t-u_{xx}=s\), \(0<x<1\), \(0<t\leq T\), subject to the initial condition \(u(x,0)=f(x)\), \(0<x<1\), to the boundary condition \(u(1,t)=g(t)\), \(0<t\leq T\), and to the mass specification condition \(\int^{b(t)}_ 0u(x,t)dx=m(t)\), \(0<b(t)<1\), for given functions \(s,f,g,b\) and \(m\). Using maximum principle arguments, the authors establish a pointwise a priori estimate for \(| u_ x(x,t)|\) yielding a similar estimate for \(\max| u(x,t)|\). Similar arguments lead to the stability estimate for the finite difference scheme providing an \(O(h^ 2+\tau)\) approximation to an equivalent problem. The influence of the input data error together with the truncation error on the finite difference solution is studied theoretically as well as numerically.
scientific article; zbMATH DE number 220563
scientific article