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Solving Stiff Ordinary Differential Equations

Ren Cang Li 

(Professors W. Kahan and J. W. Demmel)

(ARPA) DM28E04120

The best stiff ordinary differential equation (ODE) solvers are
backward differentiation formulas (BDF), each of which involves
solving a system of nonlinear equations at every step. These nonlinear 
equations may be expensive to solve. In practice, many stiff ODEs have
a common property: their solutions eventually approach some attractive 
stationary point that is often known in advance. This property has not 
been exploited in available ODE solvers. To exploit this property, we 
find various kinds of divided differences of vector functions to be 
helpful. Using them replaces nonlinear equations in the BDF by linear 
equations to be solved without iterations at each time step without 
degrading the BDFs order. Furthermore, the linear equations' solution 
tends to the known attractive stationary point as stepsize goes 
to infinity. This is crucial to maintain numerical stability at larger 
stepsizes. Whether these larger stepsizes will lead to significantly 
faster numerical methods remains to be seen.