Projected implicit Runge-Kutta methods for differential-algebraic equations. (English) Zbl 0732.65067
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.
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.
Reviewer: A.de Castro (Sevilla)
MSC:
65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
65L20 | Stability and convergence of numerical methods for ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |