Scientific programming comprises four basic elements: analyzing a physicalproblem, developing a numerical algorithm for solving the problem, designing aprogram that implements this solution within a clear and understandableframework and, finally, determining the accuracy and the limits of validity ofthe numerical solution. Before discussing these, however, we surveycomputational methods and software and hardware architecture.

Computational methods

While most textbook problems possess a high degree of symmetry and/or alimited number of variables and can therefore be solved analytically, real-worldapplications typically require numerical analysis. Further, even analyticsolutions can be unstable, as in water flowing through a cylindrical tube athigh velocity, for which the motion depends critically on the initialconditions. Numerical calculations are then performed for numerous initialconditions and statistical properties derived from the results.

Often a numerical model of a continuous system recasts the solution to the full,global problem as a set of simplified, coupled local problems, each of whichdescribes the system over a restricted spatial or temporal domain. Continuousoperators such as derivatives and integrals are then approximated by employingtheir definition as the limit of discrete differences and sums but withoutpassing to the infinitesimal limit. As an example, the response of a building toan applied force from the ground can be obtained by noting that the forces oneach brick vary only slightly over the surface of the brick. The response ofeach discrete, rigid, brick to these local applied forces can therefore easilybe evaluated. Coupling the forces and displacements on each brick to those onneighboring bricks and to those applied at the boundary between the house andthe ground generates a large system of linear equations that is solvednumerically.