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I. Discretizing the continuous physical domain into a discrete finite difference grid 2. Approximating the exact derivatives in the initial-value ODE by algebraic finite

difference approximations (FDAs) 3. Substituting the FDAs into the ODE to obtain an algebraic finite difference

equation (FDE) 4. Solving the resulting algebraic FDE

Discretizing the continuous physical domain. approximating the exact derivatives by finite difference approximations, and developing finite difference equations are discussed in this section. Brief discussions of errors and smoothness close the section.