To avoid the Ambiguity appearing in the Formulation Process using the Numerical Methods of Partial Differentrial Equations it is Important to Satisfy the following Conditions given by Fletcher C.A.J. 1989, p18 :
"The governing Equations and Auxiliary (Initial and Boundary) Conditions are Well-Posed Mathematically if the Following three Condtions are met:
- the solution exists,
- the solution is unique,
- the solution depends continuously on the auxiliary data
The above criteria are usually attributed to Hadamard[Garabedian P. 1964, p109]
There are some Flow Problems for which Multiple Solutions may be expected on Physical Grounds. These Problems would fail the above Criteria of Mathematical Well-Posedness. The Computation may be Complicated by concern about the Well-Posedness of the Mathematical Formulation
In addition we could take a simple parallel and require that for Well-Posed Computation :
- the computatinal solution exists,
- the computational solution is unique,
- the computational solution depends continuously on the approximate auxiliary data."
The Finite Difference Formulations Applied to Fluid Mechanics can be obviously Applied to Solid Mechanics.
Mohammed lamine Moussaoui