> <\body> > <\itemize> : physical/numerical. =\u> with IC and periodic BC on a Hilbert space >. :\\\>. Discretized to -dimensional space >, with projection operator >. \\> solves <\equation*> u|\t>=\\u,u(0)=\u. <\equation*> lim\>\<\|\|\>u(t)-\u(t)\<\|\|\>=0(0\t\T). <\equation*> lim\>\<\|\|\>\\(Id-\)u(t)\<\|\|\>=0(0\t\T). <\equation*> \<\|\|\>exp(\\\t)\<\|\|\>\K(0\t\T). If the above IVP is well-posed and the scheme is stable and accurate, then it converges. Look at evolution of the error: =u-\u> <\eqnarray*> |\t>e>||\\e-\\(Id-P)>>>> Integrate as ODE, estimates using accuracy and stability. : Equal to order of accuracy. <\itemize> )>: <\itemize> Semigroup property, ,t)=Id>, )||>\K*e(t-t)>>. : <\equation*> u=\x,t,|\x>u(x,t),u(x,0)=u(x) with > a polyomial of degree . (use multi-indices in -d) if > does not depend on . Then )=S(t-t)>. The above IVP is weakly well-posed (of order , with r>) if for every C> and all \0> there is a unique solution satisfying <\equation*> \<\|\|\>u(t)\<\|\|\>\C*et>\<\|\|\>f\<\|\|\>,(0\t\T). If , then . |p|>> is the Sobolev norm <\equation*> \\|\p>>u|L|>\(1+\|\\|)\|(\)\|\\. Well-posedness allows solutions by approximation. <\itemize> : <\itemize> Diagonalize systems, treat each equation separately if possible. Fourier on Jordan block: try to turn into single derivative (multiply by i\t>>?) True Jordan blocks become weakly well-posed. Unbounded eigenvalues of symbol>not well-posed. Small perturbations of a weakly well-posed symbol can make that PDE not well-posed -derivatives>: make it a system. : <\itemize> Get an energy estimate: Multiply the equation by , consider /\t E(t)>, use equation, integrate by parts to put derivatives on coefficient. <\itemize> =\(x,t,\)u+F(x,t)> also has a solution, namely <\equation*> u(x,t)=S(t,0)u(x)+S(t,\)F(x,\)d\. Proof: Differentiate solution. =\(x,t,\)v> strongly well-posed. <\equation*> u=\(x,t,\)u+\(x,t)u,u(x,0)=f(x) has solution for C>>. \\t>\<\|\|\>\(x,\)u(\)\<\|\|\>\b\<\|\|\>u(t)\<\|\|\>>>strongly well-posedness. (Proof: examine et>u(x,t)> for \0>, write down evolution, Duhamel that) Assume x=h(\t)>. |N|>> is discrete >. <\itemize> : > a vector of point evaluations, > is the shift operator in the th dimension. <\equation*> B(E,\,E)V>=B(E,\,E)V>. iff =Id>. =C(\t,\x,>,t)V>. x>> projection onto the point evaluation space. If, like in Leapfrog, we have dependency on two previous time steps: Interpret as a vector of ,V)>. Schme t)> is accurate of degree > in space and > in time> <\equation*> t>\<\|\|\>C(\t)Qx>-Qx>S(\t)u(x,t)\<\|\|\>|\>>\K(t)(\|\x\|>+\t>). For arbitrary and t=t>, <\equation*> limt\0,\x\0>(\t)Qx>-Qx>S(\t)}f(x)|N|>=0. For all , t>, <\equation*> \|C(\t)\|\K*en\t>. The difference between accuracy and convergence (which is stability) is a promise about what happens if I shrink the timestep a lot. : Plug true solution into the above. >classical solution, scheme stable>order of convergence order of accuracy, in both space and time. Proof: Write error evolution =C(\t)\+\>, write =C(\t)\>, estimate that using stability and accuracy. Can be generalized even if the IC is only in > by approximation. =C(\t)V> stable, t)\|> bounded >perturbed scheme ={C(\t)+\t*D(\t)}V> stable. Proof: =et\>V>, write down evolution for it, Duhamel that. <\itemize> Depend neither on nor . <\eqnarray*> >||(\)uu(x,0)=f(x),>>|>||(i\)(\,0)=(\).>>>> (i\)> is called the of the PDE. Weakly (strongly for ) w-p>K,\,p> independent of >: <\equation*> \|e(i\)t>\|\K(1+\<\|\|\>\\<\|\|\>)et>. Proof: Use Fourier description of Sobolev norm: (\<\|\|\>\\<\|\|\>+1)\|(\)\|\<\|\|\>>. B> for two matrices , > negative definite. \ <\equation*> \\:\(i\)+\(i\)>\\I. Proof: /\t|||>\\|||>>. (Adjoint-stuff) H(\)> hermitian with )\|,\|H(\)\|\K> <\equation*> H(\)\(i\)+\(i\)>H(\)\H(\). Proof: /\t|H(\)||>\\|||>>. (Adjoint-stuff) Remark: > is a change of variables recovering the sufficient condition. If (i\)> normal, then the IVP is well-posed iff <\equation*> Re \(\)\\. Proof: Norm coincides with the spectral radius. You only get equivalence to weak well-posedness. <\itemize> General form: <\equation*> u=A\>u,u(x,0)=u(x). <\equation*> \(i\)=i*A\. : purely imaginary eigenvalues. : <\itemize> T(\):\|T(\)\|,\|T(\)\|\K>, diagonalizes (i\)> purely imaginary eigenvalues. : weakly hyperbolic with pairwise distinct eigenvalues. : S:SAS> (!) strictly>strongly. symmetric>strongly. weakly/strongly hyperbolic>weakly/strongly well-posed\ Proof: non-normal criterion for weakly, otherwise T>. : You may invert the sign on the > without affecting strong/weak hyperbolicity. Grab a diagonalizer for >, multiply by a well-chosen diagonal matrix. <\itemize> : Use Fourier ansatz <\equation*> V=>>(k)e(j\x)> in the scheme. : <\equation*> \|V\|=>>\|(k)\|. (\t,k)> in <\equation*> =\(\t,k)(k). : <\equation*> \|{\(\t,k)}\|\K*en\t>. : Scheme stable> <\equation*> \[\(\t,k)]\e\t>=1+O(\t) VNC is sufficient if <\itemize> > is normal ((\)=||>>) or diagonalizable by a bounded and inverse-bounded diagonalizer. <\itemize> : K\G\\\n\0:\|G\|\k>. : Equivalent: <\itemize> > stable family : C> >complex 1> <\equation*> \|(A-z*Id)\|\. A\\\S\\p>> bounded, inverse-bounded s.t. > upper triangular <\equation*> \|b\|\K*min{1-\|b\|,1-\|b\|} : A\\\H\0> hermitian, bounded, inverse-bounded, <\equation*> A>H*A\H. Proof: Neumannsche Reihe, > is a change of variables for energy condition. <\itemize> of a matrix : <\equation*> \(G)=max\n>\{0}>G*V||>|||>>. normal>(G)=\(G)>. : (G)\1>>K>: ||>\K>. Proof: ||>\+(G)||>+-(G)||>>. <\itemize> of order > <\equation*> \[\(\t,k)]\1-\\|k\x\|. <\itemize> =a*u> (0>) Analytic solution: . (Left shift>Wind from right) <\itemize> preserves energy u> preserves ``mass'' \|u\|> (chop up integral at sign changes) : (Courant, Friedrichs, Lewy) <\equation*> \=at|\x>. <\equation*> =V+|2>(V-V)>> <\itemize> (2,1)-accurate (Taylor) unstable (Fourier; lin. combination of upwind and downwind scheme) <\equation*> =(V+V)+|2>(V-V)>> <\itemize> (1,1)-accurate (-e=O(\t)+O(\x)>) stable if \|\1> > error at a given point0> as t,\x\0>. (Fourier, Parseval, split tail off Fourier series) Dissipates energy: E(n)> (rewrite as )V+(1-\)V>). Dissipates mass (again, rewrite as )V+(1-\)V>) Dissipative of order 2. : <\equation*> V=V+\(V-V) <\itemize> -accurate stable for \\1> (Fourier, )=\>>, )=1-2\>, where =sin(\/2)>) : <\equation*> -V|2\t>=-V|2\x>. <\itemize> (2,2)-accurate. Stable for \1>. Not dissipative. (conserves energy) : Plug PDE into Taylor expansion of t)> until all time derivatives are gone. Use centered differences for spatial part. <\equation*> V=V+t|2\x>(V-V)+t)|2(\x)>(V-2V+V) <\itemize> (2,2)-accurate. Dissipative of order 4. : <\equation*> V=V+t|2\x>(V-V+V-V) <\itemize> (2,2)-accurate. <\itemize> =u>. =\t/\x\1/2> for standard centered difference stuff. <\itemize> =i*u>. (i\)+\(i\)>=0>>Energy conservation. centered differences are unstable. <\references> <\collection> > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||1General Framework> |.>>>>|> |math-font-series||2Well-Posedness> |.>>>>|> |2.1Lower Order Perturbations |.>>>>|> > |math-font-series||3Convergence, Stability and Accuracy> |.>>>>|> |math-font-series||4Constant Coefficient Problems> |.>>>>|> |4.1Hyperbolic Equations |.>>>>|> > |math-font-series||5Stability of Constant Coefficient Schemes> |.>>>>|> |5.1Kreiss Matrix Theorem |.>>>>|> > |5.2Lax-Wendroff Condition |.>>>>|> > |5.3Dissipative Schemes |.>>>>|> > |math-font-series||6Examples> |.>>>>|> |6.1Transport |.>>>>|> > |6.2Heat |.>>>>|> > |6.3Schrödinger |.>>>>|> >