Automated allocation of mesh points for stiff two-point boundary value problems
The numerical solution of stiff two-point boundary value problems for a system of \(n\) differential equations (1) \(dy/dt=F(t,y)\), \(a<t<b\); \(B(y(a),y(b))=0\), where \(y^ T=(y_ 1,y_ 2,\dots,y_ n)\) is a given vector, is considered. A general procedure requiring negligible prerequisites to the problem is obtained to automate the allocation of mesh points by constructing a diffeomorphism \(t=\psi(u)\) from \([a,b]\) to \([a,b]\). It is the composition of the diffeomorphisms \(t=p(s)\), whose inverse is a solution of an initial value problem (2) \(ds/dt=Q(t,y)\), \(a<t<b\); \(s(a)=c\), where \(Q>0\) is a continuous ``density function'' to be constructed, and \(s=q(u)\) being a solution of the boundary value problem (3) \(d^ 2s/du^ 2=0\), \(a<b\); \(s(a)=c\), \(s(b)=d\). A properly constructed \(\psi\) smooths out the sharp corners in the solution while it leaves the domain boundary and the boundary conditions unchanged in the augmented and transformed system (1)-(3).
scientific article; zbMATH DE number 24507