絶対値の微分方程式

コンピューターで物理的なシステムの模擬実験をする場合に微分方程式を計算しなければいけない時が多いと思います。この方程式は代数学的に解くと絶対値的に解く方法がある。機会にはほとんどの場合に代数学的な解答が無理です。
Concerns involved in solving Ordinary Differential Equations numerically include convergence, error, and generality (applicability) of the formula. Two formulas to be examined are the Adams-Bashforth method and Euler's method.
I am also confused about what constitutes a "stiff" problem which does not converge under many numerical methods.
Lastly, I am interested to determine the reason for so many libraries' having been implemented in Fortran. Is this simply due to the legacy of Fortran's use in the scientific and mathematical research communities when no other languages were available, or is there really a feature of Fortran which makes it more applicable to differential equations than C?