Numerical Approaches Differ for the 2 Basic Categories of Differential Equations
 

I. Initial-value problems.

(or "marching" probs)
 

II. Boundary-value problems

(or "jury" probs)
 
  • A. Generally hyperbolic equations. (though the definition used here is broader than the usual math definition) Generally time-dependent problems.
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  • A. Generally elliptic or parabolic equations (with some exceptions)
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  • B. Examples:
    • advection eqn
    • eqns of motion
    • hyperbolic eqns
    • many time dependent eqns (especially with propagating solutions)
  • heat eqn
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  • B. Examples:
    • Poisson eqns
    • LaPlace eqn
    • balance eqn
      (e.g. 2 = (f )
    • "omega" eqn
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  • C. Boundary Specifications Consider 2nd order wave equation with 2 2nd derivatives; it still uses just 3 boundary specifications. The idea is that the numerical soln to the equation at a later time t will dictate the "boundary condition" at t. I.E. equation is solved as initial value problem. (See figure from lecture.)

    Hence, you only need keep track of the past conditions.

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  • C. Boundary specifications Consider (2nd order) LaPlace equation. Here must specify all 4 boundary conditions in order to find the solution. (Homogeneous part determined only from boundarys.)

    Hence you need a way to bring all the boundary conditions to bear on each interior point. Must keep track of the whole domain.

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  • D. Solution procedure Usually done by retaining information in a moving "window" of time levels.
    A direct calculation (as opposed to iteration) is generally made. Iteration is of limited use since it is difficult to go back and revise an earlier estimate without redoing the whole calculation. Hence, numerical schemes tend to be sequential
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  • D. Solution procedure Iteration techniques can be used (since you must assess the influence from all the boundaries, you must go over the whole grid) Direct techniques tend to subdivide the problem: for example, multi grid method finds solutions on very coarse grids then uses those to estimate the solution on finer grids. Cyclic reduction also subdivides region
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  • E. Solution procedure concerns: You want to do each (time) step in numerical sequence as accurate as possible since you won't repeat those times. So, you look at propagation & amplitude errors and you also make sure that there are no compounding errors that lead to instability
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  • E. Solution procedure concerns: You want to make sure that your change in an estimate (when iterating) in one part of the domain does not excessively disturb the solution in another part. For direct methods, insure that errors are not magnified exeessively when subdividing or going back to higher resolution. In either case you want to avoid compounding the error that may lead to instability