The discrete second derivative is assembled as an matrix acting on the interior grid…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Arrays in memory: layout, cost and the data structures of physical problems

Problem

The discrete second derivative

\dfrac{d^2u}{dx^2}\Big|_i \approx \dfrac{u_{i-1}-2u_i+u_{i+1}}{\Delta x^2}

is assembled as an

N\times N

matrix acting on the

N=5

interior grid values, with Dirichlet boundaries (the boundary values are not among the unknowns) and grid spacing

\Delta x = 0.1\,\mathrm{m}

. ((a)) State the nonzero pattern of the matrix and compute

\mathrm{nnz}

from

\mathrm{nnz}=3N-2

. ((b)) Write the CSR arrays data, indices and indptr for this matrix, using

32

-bit integers for indices and indptr. ((c)) Using

\text{storage} = 8\,\mathrm{nnz} + 4\,\mathrm{nnz} + 4(N+1)

bytes (float64 values,

32

-bit indices), compute the total CSR storage in bytes.