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A numerical procedure for diffusion subject to the specification of mass. (English) Zbl 0773.65069

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.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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References:

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