BS methods: A new class of spline collocation BVMs

BS methods are a recently introduced class of Boundary Value Methods which is based on B-splines. They can also be interpreted as spline collocation methods. For uniform meshes, the coefficients defining the k-step BS method are just the values of the (k+1)-degree uniform B-spline and B-spline derivative at its integer active knots; for general nonuniform meshes they are computed by solving local linear systems whose dimension depends on k. An important specific feature of BS methods is the possibility to associate a spline of degree k +1 and smoothness Ck to the numerical solution produced by the k-step method of this class. Such spline collocates the differential equation at the knots, shares the convergence order with the numerical solution, and can be computed with negligible additional computational cost. Here a survey on such methods is given, presenting the general definition, the convergence and stability features, and introducing the strategy for the computation of the coefficients in the B-spline basis which define the associated spline. Finally, some related numerical results are also presented.