We know that DEAs can be converted to ODE's in index reductiom method. What is disadvantages of this procedure from the numerical point of view? What we can say about the relations between dynamic, sparsity and computational efforts of the equivalent problem and the original one?
It seems to me that your general knowledge and background of DEAs are far in excess of mine, for I had to check their definition on Wikipedia before typing this! So the following isn't an answer to your question, but simply an observation based on my experience where I had solved parabolic PDEs with an algebraic constraint.
I've used the Keller box method for solving parabolic PDEs since my PHD days. Generally such PDEs are reduced to first order form and the resulting equations are then discretised using central differences. It isn't often appreciated that this method may also be used to solve systems of ODEs - the only difference being that there are no derivatives in the "marching" direction. One can still "march", but then this will be equivalent to changing a governing parameter. In a similar way, an algebraic equation/constraint may also be added into the system. Space is too short to go into this in detail here (especially since my evening meal is almost ready!), but there is no extra complication in the Keller box methodology except that one doesn't use central differences in the marching direction, for that is no longer necessary.
Thank you very much Prof D. Andrew for your great contribution on the numerical methods for solving ODEs.
There are many methods that convert PDEs to ODEs so as to obtain the exact solution for the PDEs. Examples are Kudryshov's method, tanh methods and others.
The major disadvantage for these methods is that the are always targeting some pde not general pde problem
Jamilu: thank you for your kind words. I agree with your other statements.
The Keller box scheme is my go-to method for a large variety of problems. This includes ones with interfaces, where continuity conditions (such as for heat transport and temperature) apply. It can be used as an eigensolver, but with some more work on the basic solver. It doesn't have to be confined to systems of equations in first order form either. In my experience, the best modification to the KB methodology is to include a subroutine which evaluates numerical derivatives - thus the Jacobian matrix doesn't have to be programmed by hand. That will be a little slower than if that matrix were to be programmed by hand, but....the time needed to get a working code for a new problem is reduced very very considerably, and makes some rather large problems easily achieveable.
Differential equations are normally derived on the basis of a rate of change (velocity) or a 2nd derivative (acceleration) that give a relation based on a physical law such as F=m*a. Expressing the force as a function of time allows one to solve for the acceleration, integrate that to get velocity, and integrate that to get position. Some differential equations are not as well-behaved, and show singularities due to a failure to model the problem correctly, or a limitation of the model that was not apparent. Some DE’s can be solved analytically in closed-form, but most have to be approximated by numerical procedures, which can be unstable.
Since an ordinary differential equation involves a derivative over a single variable, usually in a univariate context thus, it has a disadvantage to convert to it.
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