Our system of nonlinear algebraic equations could very well contain 2.2 billion variables.
2.
Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of ( usually ) nonlinear algebraic equations.
3.
If the partial differential equation is nonlinear, the discretization will also be nonlinear so that advancing in time will involve the solution of a system of nonlinear algebraic equations, though linearizations are possible.
4.
The Laplace transform could in principle be applied to PDEs involve higher derivatives, but, it has the disadvantage that involves 0, and also it requires the analysis of high order nonlinear algebraic equations.
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