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
(x-ut) equal to a constant. 

      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}