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Extra info for Generalized difference methods for differential equations. numerical analysis of finite volume methods

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A fundamentally different aspect comes into play by adopting the viewpoint of differential equations on manifolds, as introduced by Rheinboldt [Rhe84]. 48b) of index 2 where the constraint 0 = b(y), assuming sufficient differentiability, defines the manifold M := y ∈ Rny : b(y) = 0 . 49) for the matrix product ∂b/∂y · ∂a/∂z implies that the Jacobian B(y) = ∂b(y)/∂y ∈ Rnz ×ny possesses also full rank nz . 55) spans the kernel of B and has the same dimension ny − nz as the manifold M. 48b), which, starting from a consistent initial value, is required to proceed on the manifold.

36b) is consistent if the constraint is satisfied, 0 = b(y 0 , z0 ). 49) in a neighborhood of the solution. 50) and 0 = B(y) ∂a d ∂a (y, z)˙z + B(y) (y, z)a(y, z) + B(y) · a(y, z). 48b) is thus k = 2. As a rule of thumb, differential-algebraic equations of index 2 or higher are generally more difficult to analyze and to solve numerically than ordinary differential equations or DAEs of index 1. 50). , 0 = b(y 0 ), 0 = B(y 0 )a(y 0 , z0 ). 52) In practice, finding such consistent initial values may constitute a challenging problem of its own [BCP96, ST00].

34). An exception is the case k = 1 where N 1 is the zero matrix. 34) in a stable way, the computation of the Jordan canonical form and consequently also of the Kronecker form are notoriously ill-conditioned problems. 40), we introduce new variables and right hand side vectors V −1 x =: y , z U c =: δ θ . 43a) N z˙ + z = θ . 43a) follows by integrating and results in an expression based on the matrix exponential exp(−C(t − t0 )), Eq. 43b) for z can be solved recursively by differentiating. More precisely, we have N z¨ + z˙ = θ˙ ⇒ N 2 z¨ = −N z˙ + N θ˙ = z − θ + N θ˙ .

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