FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL …

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FINITE DIFFERENCE METHODSFORSOLVING DIFFERENTIAL EQUATIONSI-Liang ChernDepartment of MathematicsNational Taiwan UniversityMay 16, 20132Contents1 FINITE DIFFERENCE Approximation . . . . . . . . . . . . . . . . . . .. . . . . . . . Basic Numerical METHODS for Ordinary DIFFERENTIAL Equations . . . . . . . . . . . Runge-Kutta METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . Multistep METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Linear DIFFERENCE equation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . Zero Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182 FINITE DIFFERENCE METHODS for Linear Parabolic FINITE DIFFERENCE METHODS for the Heat Equation.

solutions generated by the Euler method form a Cauchy sequence. Backward Euler method In many applications, the system is relaxed to a stable solution in a very short period of time. For instance, consider y′ = y¯−y τ. The corresponding solution y(t) →y¯as t∼O(τ). In the above forward Euler method, practically, we should require 1 ...

  Euler, Cauchy