|
|
|
Propagation of a cosine profile
|
|
|
To begin with, we consider a simple one-dimensional example that allows comparing
the performance of different time-stepping algorithms.
Consider a transient pure convection problem with initial condition
with x0 = 0,2
and s = 0,12.
The problem is solved using a uniform mesh of linear elements.
Several integration schemes are available: three explicit
 |
second-order Lax-Wendroff finite element method (consistent mass representation) |
| |
second-order Lax-Wendroff finite element method with diagonal mass representation
|
| |
third-order explicit Taylor-Galerkin method
|
and three implicit
| |
second-order Crank-Nicolson finite element method (consistent mass representation) |
| |
second-order Crank-Nicolson finite element method with diagonal mass representation
|
| |
fourth-order implicit Taylor-Galerkin method
|
Some results
Figures below show solution at time t=0.6 obtained using different methods with
Courant number C=0.9.
|
| |
|
|
|
|
|
| Lax-Wendroff |
|
Lax Wendroff with
lumped mass matrix |
|
Third order
Taylor Galerkin |
|
| |
|
|
|
|
|
| Crank-Nicolson |
|
Crank-Nicolson with
lumped mass matrix |
|
Fourth order
Taylor Galerkin |
Why do we use the lumped mass matrix?
The two figures below show solution at t = 0.15 obtained with Lax-Wendroff method
using the consistent mass matrix (left) and the lumped one (right).
Note that, as Courant number is the same in both cases, stability range increases
when using a lumped mass matrix representation.
However, when keeping C in the stability range, results are more accurate
if the consistent mass matrix representation is used
|
|
Back to Unsteady Transport 