Department of Mathematics Numerical Solutions to Partial Differential Equations Ch 12. Applied mathematics. Korea University.

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Presentation transcript:

Department of Mathematics Numerical Solutions to Partial Differential Equations Ch 12. Applied mathematics. Korea University.

Department of Mathematics Elliptic Partial Differential Equations Ch12.1 Index Parabolic Partial Differential Equations Ch 12.2 Hyperbolic Partial Differential Equations Ch 12.3 An Introduction to the Finite-Element Mothod Ch 12.4

Department of Mathematics  Elliptic Partial Differential Equations : poisson equation  Parabolic Partial Differential Equations : Heat, diffusion equation  Hyperbolic Partial Differential Equations : wave equation

Department of Mathematics Ch 12.1 Elliptic Partial Differential Eqeations  Poisson equation

Department of Mathematics Using Taylor series in the variable about

Department of Mathematics Poisson equation at the points Boundary condition

Department of Mathematics Finite – Difference method with truncation error of order Boundary condition

Department of Mathematics

Ch 12.2 Parabolic Partial Differential Eqeations  Parabolic partial differential equation : boundary condition : initial condition

Department of Mathematics Using Taylor series in

Department of Mathematics boundary condition initial condition : local truncation error

Department of Mathematics let : initial condition : Forward Difference method

Department of Mathematics is made in representing the initial data If At n-th time step the error in is. The method is stable The Forward Difference method is therefore stable only if

Department of Mathematics : eigenvalues of A or The Forward Difference method is conditionally stable with rate of convergence

Department of Mathematics To obtain a method that is unconditionally stable : Backward-Difference method where

Department of Mathematics The matrix representation

Department of Mathematics : eigenvalues of A At n-th time step the error in is. The Backward-Difference method is unconditionally stable method. The local truncation error for method is of order. Richardson’s method

Department of Mathematics Crank-Nicolson method : Forward-Difference method at j-th step in : local truncation error : Backward-Difference method at (j+1)th step in : local truncation error Assume that average – difference method

Department of Mathematics The matrix representation where

Department of Mathematics Ch 12.3 Hyperbolic Partial Differential Eqeations  Hyperbolic partial differential equation

Department of Mathematics Using centered-difference quotient

Department of Mathematics

Ch 12.4 An Introduction to Finite- Element Method boundary condition

Department of Mathematics Polynomials of linear type in and

Department of Mathematics

linear system

Department of Mathematics