Discontinuous Galerkin methods Lecture 1 - Brown University

Finite difference methods
roduction
neighborhood of each grid point xk 1 Introduction
, the solution and the flux are as
d by local polynomials
• The local approximation is a 1D polynomial
• The equation is satisfied in a pointwise manner
t, in the neighborhood of each grid point xk , the solution and the flux are assumed to be w
roximated by local polynomials
1 Introduction
own in space and spatial derivatives are approximated by difference metho
at is, the conservation law is approximated as
duh(xk ,t)
+
dt
fh(xk+1 ,t) − fh(xk−1 ,t)
hk + hk−1 = g(x k ∈ [x
,t), (1
k−1 ,x k+1 2
]: uh(x, t) = ai(t)(x − x
i=0
k ) i 2
, fh(x, t) = bi(t)(x −
i=0
efficients ai(t) and bi(t) are found by requiring that the approximat
he grid points, xk x ∈ [x
. Inserting these local approximations into Eq.(1.1
k−1 ,x k+1 2
]: uh(x, t) = ai(t)(x − x
i=0
k ) i 2
, fh(x, t) = bi(t)(x − x
i=0
k ) i ,
re the coefficients ai(t) and bi(t) are found by requiring that the approximate function in
ates at the grid points, xk . Inserting these local approximations into Eq.(1.1), results in
dual
x ∈ [x k−1 ,x k+1 ]: Rh(x, t) = ∂uh + ∂fh − g(x, t).