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The usual approach to solving these kinds of system is not to actually think of them as an ODE with an algebraic constraint, but to use specialized algorithms that makes use of the structure. The prototypical example from which you can learn is the Navier-Stokes equations, in which you have one equation that contains the time derivative of the velocity, and ...
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Finite element methods are generally easier to deal with when using adaptive mesh refinement because higher order finite difference methods have stencils that extend for several mesh sizes away from a point, and one has to deal with all of the possible hanging nodes one might encounter in this large neighborhood even if the mesh is rectangular and is derived ...
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Since, the title of your question has a term 'finite difference' into it, I assume you are looking for a software for finite difference method. Finite difference methods are restrictive. Usually softwares are made for wide situations and range of the problems. Finite volume and finite element methods offer greater flexibility than finite difference methods. ...
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