Goals : 1) Practice numerical methods to solve simple PDEs with finite differenc

Question

Goals: 1) Practice numerical methods to solve simple PDEs with finite difference equations; 2) Understand the linear computational stability.

Problem # 1

Integrate the linear wave equation using values typical of large-scale models. You can write your own MATLAB program or modify a sample program I offered to you (see the attachment).

?u = ?c ?u

?t ?x Boundary conditions: periodic

Initial conditions:

u(x,0) = c + Asin(kx)
c = 20ms?1, A =10ms?1,?x = 200km,k = 2? / L with_ L = 10?x

(a) Use leapfrog scheme. Choose two time steps: one satisfies the CFL condition and the other violates it. 1) How long does it take to “blow up”; 2) Calculate the total kinetic energy of the solution u (0.5 u2 ) at each time step; write a simple MATLAB (or IDL) program and draw a curve to show the variation of total kinetic energy as function of the number of time steps.

(b) Compare with the exact solution, computer the root-mean-square (RMS) error R and the relative error RE. Then,
Repeat with A=25 m/s
Repeat with L = 4Dx .

Prepare a table to summarize R and RE.

(c) Modify the programs; use the upstream scheme (instead of leapfrog scheme) and repeat (a)- (b).

Lab Report:

Based on your results to solve above problem, write a report (minimum 1 page of text for each problem; 12pt Times New Roman; doubled space) to discuss the linear computational stability problem. You can attach figures to support your conclusion.

codes for matlab:

Explanation / Answer

Order of a PDE: The order of the highest derivative term in the equation is called the order of the PDE. Thus equations (6.1.1 to 6.1.6) are all of second order.

Linear PDE: If the dependent variable and all its partial derivatives occure linearly in any PDE then such an equation is called linear PDE otherwise a non-linear PDE. In the above example equations 6.1.1, 6.1.2, 6.1.3 & 6.1.4 are linear whereas 6.1.5 & 6.1.6 are non-linear.

Quasi-linear PDE: A PDE is called as a quasi-linear if all the terms with highest order derivatives of dependent variables occur linearly, that is the coefficients of such terms are functions of only lower order derivatives of the dependent variables. However, terms with lower order derivatives can occur in any manner. Equation 6.1.5 in the above list is a Quasi-linear equation.

Homogeneous PDE: If all the terms of a PDE contains the dependent variable or its partial derivatives then such a PDE is called non-homogeneous partial differential equation or homogeneous otherwise. In the above six examples eqn 6.1.6 is non-homogeneous where as the first five equations are homogeneous.

Notation: It is also a common practise to use subscript notation in writing partial differential equations. For example the Laplace Equation in three dimensional space

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