A Finite Element Scheme for Shock Capturing Part 3 ppsx

A

where C is a constant determined by the boundary condition and p is the

numerical root.

The roots of Equation 36 are

which makes the general solution

where b is some constant.

The analytic solution corresponds to p = 1. The spurious node-to-node

oscillation is the root p = -1. This root results from a test function which is

made up solely of even functions; that is, the test function, the hat function, is

symmetric about node i (Figure 5). If we consider the node-to-node oscilla￾tion, its derivative is an odd function, the inner product of which with the test

function is identically zero. This is a solution!

Now, if the test function includes both odd and even components, this

mode will no longer be a solution. In fact, if we weight the test function

upstream, these oscillations are damped; weighting downstream amplifies them.

A common approach is to use a test function, q, weighted as follows,

where a is a weighting parameter.

Here the spatial derivative supplies the odd component to the test function.

The resulting discrete solution using this test function is

from which the numerical roots may be calculated by

Chapter 2 Numerical Approach