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FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL …

FINITE DIFFERENCE METHODSFORSOLVING DIFFERENTIAL EQUATIONSI-Liang ChernDepartment of MathematicsNational Taiwan UniversityMay 16, 20132 Contents1 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 .. Some discretization METHODS .. Stability and Convergence for the Forward euler method .. von Neumann Analysis .. Energy method .. Stability Analysis for Montone Operators Entropy Estimates .. Entropy estimate for backward euler method .. Existence Theory .. Existence via forward euler method .. A Sharper Energy Estimate for backward euler method.

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 ...

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