Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.

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Presentation transcript:

Finite Difference Methods Definitions

Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation is the truncation error

Finite Difference Methods Given a smooth function (i.e., a function that is continuous and has continuous derivatives), the Taylor series approximates the value of a function at one point based on the value of the function and its derivatives at another, nearby point

Finite Difference Methods First-order Taylor series: solving for the first derivative

Finite Difference Methods Truncation error:

Finite Difference Methods Exercise: write the Taylor series expansion for q(x -  x) and solve for the first differential...

Finite Difference Methods x t ii - 1i + 1 n - 1 n n + 1 xx tt

Finite Difference Methods Backward time difference: explicit Forward time difference: implicit