This paper considers differential-algebraic equations (DAE) \(x'=f(x,y,t)\) with a constraint \(g(x,t)=0,\) where \[ (\partial g/\partial x)(\partial f/\partial y) \] is assumed to be nonsingular for all t,x,y in a neighborhood of the solution, and an implicit Runge-Kutta (RK) method (*) \(X'_ i=f(X_ i,Y_ i,t_ i), g(X_ i,t_ i)=0, (i=1,2,...,k)\), \(x^*_ n=x^*_{n-1}+h_ n\sum^{k}_{j=1}b_ jX'_ j,\) where \(X_ i=x^*_{n-1}+h_ n\sum^{k}_{j=1}a_{ij}X'_ j,\sum^{k}_{j=1}a_{ij}=c_ i, t_ i=t_{n-1}+h_ nc_ i.\)
These implicit RK methods can present, when applied to DAEs, order reduction and some problems of instability. This paper presents a new class of methods, projected implicit RK methods, defined by putting in (*) \(x^*_{n-1}=x_{n-1};\) let \(x_ n=x^*_ n+\lambda_ n(\partial f/\partial y)(x_ n,y_ n,t_ n), \lambda_ n\) determined by \(g(x_ n,t_ n)=0.\)
The main application of these methods seems to be the improvement of methods for boundary value problems based on symmetric discretizations. Existence, stability and convergence for the linear case is proved and applied to other problems.