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Large discrete PDE in sections
When using accurate discrete computation of derivatives,
the accuracy degrades at about 13 points used in computation of
the derivative. Thus there is a need to compute the discrete
derivatives in sections, sometimes referred to as parts.
For background see Numerical Differentiation
and many equations for numerical computation of discrete derivatives
The case of interest in this lecture is the following.
We gave a differential equation in one variable d^4U(x)/dx^4 = f(x)
or ∇^2 U = f(x) or &del;^4 U = f(x)
We want the values of U(x) at x = 0, 1, 2, ..., 6
If we only use the minimum order discrete derivative, 5, from nderiv.out
we have the following 7 equations in 7 unknowns:
Examples of using sections are shown below.
pde22a_eqp.java source code
pde22a_eqp_java.out solution
pde22a_eqp_java.sh for plot
pde22a_eqp_java.plot for plot
rderiv.java derivative source code
simeq.java simultaneous equation source code
Compiled and executed using:
javac -cp . rderiv.java
javac -cp . simeq.java
javac -cp . pde22a_eqp.java
java -cp . pde22a_eqp > pde22a_eqp_java.out
./pde22a_eqp_java.sh # gnuplot
gimp pde22a_eqp_java.png
The plotted output solution is:
For comparison with use of single section for discrete derivative
pde22a_eq.java source code
The one dimensional sections started from
pde11p_eq.java source code
using solve_pde function with single section or part
then added conversions to sections using solve_pdep
with output
pde11p_eq_java.out output
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