Tomas Vejchodsky,
Higher-order discrete maximum principle for 1D diffusion-reaction problems,
submitted to Appl. Numer. Math., 2008

[Download preprint]     [Download the matlab codes]     [Homepage of INTLAB]

Verification of the assumption.

On this page we present matlab codes for verification of the assumption (62) from the above paper using the interval arithmetic. In particular, we verify nonnegativity of a rational function ωp(θ,γpθ+δp,ξ,η) for θ∈(0,1/2], ξ∈[-1,1], η∈[-1,1] and for p=3,4,...,10. This function is defined in the paper by formulas (56)–(57), where also the following relations are needed: (42), (40), (26), (20)–(22), (18), and the definition of the generalized eigenvalues and eigenfunctions of the Laplacian on page 6. The values of constants γp and δp are listed in Table 1.

To run the matlab codes the package INTLAB is needed. The archive DMPabs_matlab.tar.gz with the matlab codes contains the following files:

testpall.m
Run the test for p=3,4,...,10.
nntest_vf.m
The core routine. It checks nonnegativity of a function of three variables on a rectangular cuboid.
Ghp_p3.m, ..., Ghp_p10.m
Definitions of the function ωp(θ,γpθ+δp,ξ,η) for p=3,4,...,10.

The results of the computations are summarized in the following table.
polynomial
degree p		 nonnegativity
verified		 elapsed time maximal level
of recursion		 number of interval
function evaluations		 log file
3 YES 13.452671 sec. 4 3837952 nntest_p3_log.tar.gz [63 KB]
4 YES 673.328425 sec. 5 93155328 nntest_p4_log.tar.gz [1.2 MB]
5 YES 3315.276273 sec. 6 321048576 nntest_p5_log.tar.gz [3.0 MB]
6 YES 6700.862031 sec. 5 593965056 nntest_p6_log.tar.gz [4.9 MB]
7 YES 41282.330112 sec. 6 2.866721e+09 nntest_p7_log.tar.gz [17 MB]
8 YES 265155.368104 sec. 7 1.403913e+10 nntest_p8_log.tar.gz [80 MB]
9 YES 365928.754777 sec. 7 1.545936e+10 nntest_p9_log.tar.gz [75 MB]
10 YES 1198768.53791 sec. 6 4.888682e+10 nntest_p10_log.tar.gz [210 MB]

Last actualization 2008-12-09.
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