Problem Statement A.
Peclet problem DE with BCs for the 1-D steady energy transport in non-dimensional form are
DE:
BCs: q(0)
= 0, q(1) = 1 The analytical solution for this “Peclet” problem statement is
which exhibits a pronounced
wall-layer appearance as Pe increases.
The Galerkin weak statement formulation leads to the template
pseudo-code GWSh = S{WSe} = {0} {WSe} = (PEI) ( ) { } (-1) [A211] {Q}
+ ( ) ( ) {U} ( )[A3001] {Q} The theoretical asymptotic
error estimate for this GWSh algorithm is
This Peclet problem requires use of solution-adapted non-uniform meshings. A uniform distribution of non-uniformity results from the use of the geometric progression ratio formulation
For Pr > 1, the meshing
progressively coarsens, while Pr < 1 results in generation of progressive
mesh refinement. B. Traveling wave problem
DE and BCs for 1-D unsteady pure advection mass transport are
DE:
BCs: q(x,
t=to) = qo(x) The analytical
(characteristic) solution for this problem statement is q(x,
t) = qoexp(x-ut) which confirms that the
initial distribution qo(x) appears unchanged for
all values of The GWSh process leads to the time-dependent ODE system.
GWSh Ž
Inserting this into the time Taylor series
yields
the resultant GWSh + qTS matrix statement and solution update are
([MASS]+qDt[UVEL]){DQ}
= -Dt[UVEL]{Q}n
Recalling the template syntax
{WSe} = (const)(avg.data)e{dist.data}e(det)[FE
matrix]{Q or data}e The resultant
algorithm component pseudo-code statements are JACe = ( ) ( ) { } (1) [A200] { } + (qDt) ( ) {U}( ) [A3001] { }
DtRESe
= (Dt)
( ) {U}( ) [A3001] {Q}
The theoretical asymptotic error estimate for this GWSh
+ qTS
algorithm is
Hence the solution
convergence rate is independent of basis degree k. Solution quality
is also dependent on mesh resolution of the IC qo, along
with q
and Dt.
A parameter conveniently combining these data is the non-D time step, called
the “Courant Number”, with definition
For a uniform meshing le
is a constant, hence dividing through by le the
alternative form for the template pseudo-code specifically dependent on
Courant number is JACe = ( ) ( ) { } (0) [A200] { } + (q) ( ) {C} (0) [A3001] { } Dt×RESe = ( ) ( ) {C} (0) [A3001] {Q} |