General strategy for solutions of computational problems

Definition

A basic general strategy is to replace the complex problem with a simpler problem that also yields the same solution (or atleast a closely related solution). These are some examples:

1. Replace infinite-dimensional spaces with finite-dimensional spaces
2. Replacing infinite processes with finite processes (ie. changing integral to finite summation)
3. Replacing differential equations with algebraic equations
4. Replacing nonlinear problems with linear problems
5. Replacing high-order systems with low-order systems (highest derivative of dependent variable)
6. Replacing complicated functions with simpler functions (like polynomials)
7. Replacing general matrices with matrices that have a simpler form

Most solutions using this strategy are approximate solutions, so the primary goal afterwards is to find the accuracy of the approximate solution and establish converge to the real solution in the limit.

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Example

For example, to solve a system of nonlinear differential equations, we might first replace it with a system of nonlinear algebraic equations.

Then replace the nonlinear algebraic system with a linear algebraic system.

Then replace the matrix of the linear system with one of a special form for which the solution is easy to compute.

At each step of this process, we would need to verify that the solution is unchanged, or is at least within some required tolerance of the true solution.

References

"What are high/low - order systems?" - Higher Order Linear Differential Equations