间断伽辽金
\emphWhat is DG？ An Example
consider [
Standard Finite Element Method:
\item Approximate: u(x)=∑jujϕj(x),where ϕj are ‘hat’ basis function.
\item PicTUre
\item Weak form of equations:\
substitute u(x)=∑jujϕj(x),our goal is solve for the uj\
−∑juj∫Iϕjϕi,xdx+u4ϕi−u0ϕi=∫Icos(x)ϕidx
Where DG differ
\begin{enumerate}
\item Formulation is the same as standard FEM;
\item DG difference is the choice of basis,no continuity constraint between elements;PICTURE
\item Numerical flux
\item DG weak form:
$$\int_{I_k} u_x \phi_i dx =\int_{I_k} cos(x)\phi_i dx $$\
$$-\int_{I_k}u\phi_{i,x}dx +u\phi_i \mid {x^-{k-\frac{1}{2}}} ^{x^-{k+\frac{1}{2}}}=\int{i_k} cos(x)\phi_i dx
\emphDG for conservation law\
[
\item Approximate:
uh(x,t)=∑l=0ku(l)i(t)v(i)l(x) for x∈Ii where vi0(x)=1,vi1(x)=x−xiΔxi/2,vi2(x)=(x−xiΔxi/2)2−13 is the orthogonal basis over Ii; u(l)i(t)=1∫Ii(v(i)l)2dx∫Iiuh(x,t)v(i)l(x)dx are degree of freedom\item DG weak formulation ∫Iiutvdx−∫Iif(u)vxdx+f(u(xi+1/2,t))v(xi+1/2)−f(u(xi−1/2,t))v(xi−1/2)=0 (it’s not Ok! )
\item Numerical flux f^i+1/2=f^(u−i+1/2,u+i+1/2) to replace the f(ui+1/2) \item[1] monotone $\hat{f}(\uparrow,\downarrow)$\item[2] $\hat{f}$ is Lipschitz continuos with respect to both arguments.\item[3] $\hat{f}(u,u)=f(u)$\end{description}
\item Example of numerical flux
\begin{description}
\item[Lax-Friederich]$$\hat{f}^{LF}(a,b)=\frac{1}{2}[f(a)+f(b)-\alpha(b-a)],\alpha=max|f’(u)|$$
Particular :if $f(u)=u\Rightarrow \hat{f}=\hat{f}(u^-{i+1/2},u^-{i-1/2})=u^-_{i+1/2}$
\item[Godunov] [\hat{f}^G(a,b)=\begin{cases}
\underset{a \leqslant u\leqslant b}{\mathrm{min}} f(u)\ if\ &a\leqslant b\
\underset{a > u > b}{\mathrm{max}} f(u) \ if \ &a> b
\end{cases}]
\item Semi-discretization scheme
ddtu(l)i+1al(−∫Iif(uh)ddxv(i)l(x)dx+f^i+1/2v(i)l(xi+1/2)−f^i−1/2v(i)l(xi−1/2))=0we write the above as ut=L(u)
\item Time discretization(Runge-Kutta)[Shu and Osher(1989)]
u(1)u(2)un+1=un+ΔtL(un)=34un+14u(1)+14ΔtL(u(1))=13un+23u2+23ΔtL(u(2))
\end{enumerate}
\boxed{\emph{\textbf{Cell entropy inequality and L2 -stability}}}
\begin{description}
\item[entropy condition] $$U(u)_t+F(u)_x \leqslant 0$$
where $U(u)$ is entropy with $U”(u)\geqslant 0$ and the corresponding entropy flux $F(u)=\int ^u U’(u)f’(u)u_x dx$\
\item[Proposition (Jiang and Shu 1994)]:\
(1) The solution $u_h$ to the semi-discrete DG scheme satisfies the following cell entropy inequality
$$\frac{d}{dt}\int_{I_j} U(u_h)dx +\hat{F} {j+1/2} -\hat{F}{j-1/2}\leqslant 0$$ for the square entropy $U(u) = u^2/2$ and for some consistent entropy flux $\hat{F} {j+1/2}=\hat{F}(u^-{h,j+1/2},u^+_{h,j+1/2}) ,with\hat{F}(u,u)=F(u)=uf(u)-\int ^u f(u)du (2)Furthermore,uhsatisfiesthefollowingL^2 −stability ddt∫10(uh)2dx⩽0 $|u_h(\cdot,t)|{L^2}\leqslant |u_h(\cdot,0)|{L^2}\leqslant |u_0|_{L^2}$$
\emphLimiter and total variation stability
\item[Requirement on the limiter]:\
\item Maintain the local conservation: keep the cell average;\item Do not degrade the accuracy of the scheme\end{itemize}\item[PIcture1,2]
\item[Minmod limiter]
with [m(a_1,a_2..a_l)= {s⋅min(|a1|,|a2|,..|al|) ifs=sign(a1)=sign(a2)=..=sign(al)0 other ]
Then set u(mod)h(x−i+1/2)=u¯i+u~(mod)i , u(mod)h(x+i+1/2)=u¯i+u~~(mod)i
New uh=\
\item[ The solution before (solid line) and after (dashed line) using the limiter: Pictur]
\item[Lemma (Harten 1997)] If a scheme can be written in the form un+1i=uni+Ci+1/2Δ+uni−Di−1/2Δ−uni periodic or compacted supported boundary conditions, where Ci+1/2 and Di?1/2 may be nonlinear functions of the grid values unj for j=i−p,....,i+q with some p,q⩾0, satisfying Ci+1/2⩾0,Di+1/2⩾0,Ci+1/2+Di+1/2⩽1
then the scheme is TVD,namely: TV(un+1)⩽TV(un) where TV(u)=∑i|Δ+ui|
\item[Proposition:] The solution un h of the DG scheme with the forward Euler time discretization using the limiter discussed above, is total
variation diminishing in the means (TVDM), that is TVM(un+1h)⩽TVM(unh) .with the semi-norm defined as TVM(uh)=∑i|Δ+u¯i|.
Similar result can be extended to high order SSP time discretizations
\item[How about accuracy]?
\
\item In the smooth, monotone region:assume uh is an approximation to a locally smooth function u, thenu~i=12ux(xi)Δxi+o(h2),u~~i=12ux(xi)Δxi+o(h2)
while:
\item At the smooth extrema:Accuracy loss!PICTURE
Δ+ui¯Δ−ui¯<0,and u~(mod)i=m(ui¯,Δ+ui¯,Δ−ui¯)=0
\item[\textbf{TVB limiter}]:\
[\tilde{m}(a_1,a_2..a_l)= {a1 if|a1|<Mh2m(a1,...al) other ]
Errorestimate:
\item[\textbf{Proposition}]Let u be the smooth exact solution to the conservation law ut+ux=0, and let uh be the numerical solution to the semi-discrete DG method, then ∥u−uh∥L2⩽Chk+1 here the constant C depends on the exact solution and it is independent of h<script type="math/tex" id="MathJax-Element-1204">h</script>,
总结
