Debunking the Myth: Implicit ODE Solvers Are Not Universally Superior to Explicit Methods
By
cbolton
Pure flour-power. Hearty enough to carry you through lunch.
Summary
This article challenges the common belief that implicit ODE solvers are universally more robust than explicit methods. It argues that the choice between implicit and explicit solvers depends on the specific mathematical properties of the differential equation being solved, with neither approach being universally superior. The author explains that implicit methods excel on stiff problems but can be inefficient or problematic for certain equation types, while explicit methods have their own advantages in specific contexts.
Key quotes
· 3 pulledA very common adage in ODE solvers is that if you run into trouble with an explicit method, usually some explicit Runge-Kutta method like RK4, then you should try an implicit method.
Implicit methods, because they are doing more work, solving an implicit system via a Newton method having 'better' stability, should be the thing you go to on the 'hard' problems.
This turns out to not be true, and really understanding the ODEs will help us understand better why no ODE solver is best.
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