The WENO method. Solution proposal to Project 2: MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven

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1 MATH-459 Nmerical Metods for Conservation Laws by Prof. Jan S. Hestaven Soltion proposal to Project : Te WENO metod Qestion. (a) See SHU 998. (b) In te ENO metod for reconstrction of cell bondaries we sp a lot of comptational eort for eac cell to create and compare varios stencils of lengt K before nally coosing one of tem. Te idea wit te WENO metod is, instead of sping so mc eort comparing stencils and coosing te most appropriate, lets jst take all of tem and instead sp comptational eort on calclating some set of weigts to be sed at eac cell, tis as te added advantage of leading to O ( d k ) accrate approimations in smoot regions. Qestion. (a)(b) See DEMO script and fnctions attaced. (c) Te initial conditions are depicted in grer. Te soltion for eac initial condition at T= is presented in gre for K = and in gre 3 for K = 3. We notice tat for K = everyting looks nice and smoot bt for K = 3 in te case of sock initial condition oscillations appear. Wen sing a La-Friedrics type, te large nmerical dision dampens tese oscillations, wen sing te Godnov's type wit Roe's approimate Riemann solver, tere is less articial dision and te oscillations appear larger. (d) Te rst initial condition is smoot and remains smoot trogot te soltion, te second initial condition is also smoot bt two discontinities arise dring te soltion procedre. Te tird initial condition contains two socks and trogot te soltion to tis initial conditions discontinities are presented. (e) See DEMO script and fnctions attaced and grer 4 containing te accracy reslts. (f ) For te case of te smoot initial condition, initial condition, we measre tird order accracy. For te two initial conditions, and 3, containing discontinities, te accracy degenerates to rst order. (g) Yes. (f ) First order. (i) Also rst order, bt te error constant is sbstantially better ten wen solving initial condition 3. Qestion.3 (a) Done in previos project. (b) See DEMO script and fnctions attaced. (c) See grer 4 containing te accracy reslts (d) Wen sing WENO reconstrction, te dierence in accracy between sing a la-friedric's type and sing a Godnov's type wit roes approimate Riemann matri is qalitatively te same as wen simply approimating cell interfaces wit cell averages. Tere is an error constant dierence between te La-Friedrics and te Godnov's, te Godnov's is sbject to less nmerical dision tan te La-Friedrics type.

2 (a) Initial condition (b) Initial condition (c) Initial condition 3 Figre : Te initial conditions q(, ).

3 (a) La-Friedrics and initial data set..5 (b) Roe's metod and initial data set (c) La-Friedrics and initial data set..5 (d) Roe's metod and initial data set (e) La-Friedrics and initial data set 3..5 (f) Roe's metod and initial data set 3. Figre : Soltion at T = sing =.5 wit K =. 3

4 (a) La-Friedrics and initial data set..5 (b) Roe's metod and initial data set (c) La-Friedrics and initial data set..5 (d) Roe's metod and initial data set (e) La-Friedrics and initial data set 3..5 (f) Roe's metod and initial data set 3. Figre 3: Soltion at T = sing =.5 wit K = 3. 4

5 La-Friedrics Roes-Metod O(d3) Error (a) Cells La-Friedrics Roes-Metod O(d) Error (b) Cells La-Friedrics Roes-Metod O(d) Error (c) Figre 4: Accracy as measred in te -norm against a Roe's metod soltion sing grid cells wit K = on initial condition (a), (b), (c) 3. Cells 5

6 clear all,clc % In tis DEMO script for project we attempt to solve te one dimensional sallow % water eqation. Te problem can be formlated as a conservatioin law in te form % of a system of two copled nonlinear partial differential eqations. Te soltion % can be visalized and te accracy of te sed metod can be measred. % Visalize and estimate accracy Visalize = ; Accracy = ; % Nmerical fl fnctions available types = {'La Friedrics','Roes Metod'}; % Pysical constants g = ; % Coose stencil size to se K = (3) or K = 3(5) K = ; C = GetC(K); % Coose initial dataset to 3 data = ; % Do magic if Visalize view = [ ]; =.5; [k,x,nx,t,nt] = Setp(); for i = :nmel(types) Q = InitialData(data,X,nX,/); F = zeros(,nx+); figre(i); plot(x,q(,:),' b',x,q(,:)./q(,:),' r'); ais(view); grid on; label('');leg('',''); %matlabtikz([nmstr(data),'_initial.tikz'], 'eigt', '\figreeigt', 'widt', '\figrewidt'); Qedge = zeros(,*nx+); for t = :nt Q = SSPRK3(F,C,,nX,X,Q,K,g,k,types{i}); plot(x,q(,:),' b',x,q(,:)./q(,:),' r'); ais(view) drawnow [Q,X] = GetRef(data,g,K); Qplot(,:) = interp(x,q(,:),x); Qplot(,:) = interp(x,q(,:),x); old on;plot(x,qplot(,:),'k',x,qplot(,:)./qplot(,:),'k'); label('');leg('','','');grid on; name = [types{i},'d',nmstr(data),'k',nmstr(k),'_soltion.tikz']; matlabtikz(name,'eigt', '\figreeigt', 'widt', '\figrewidt'); if Accracy % Calclate soltion [Q,X] = GetRef(data,g,K); % Measre accracy of metods figre(nmel(types)+); n =.^(:.:.); =./ceil(n); error = zeros(nmel(),); cells = zeros(nmel(),); slope = zeros(nmel(types),); legen = {' ob',' or',' k'}; for i = :nmel(types) for j = :nmel() [k,x,nx,t,nt] = Setp((j)); 6

7 Q = InitialData(data,X,nX,(j)/); F = zeros(,nx+); for t = :nt Q = SSPRK3(F,C,(j),nX,X,Q,K,g,k,types{i}); % Measre error error(j) = norm(interp(x,q(,:),x) Q(,:),)/nX; cells(j) = nx; % Plot slope(i,:) = polyfit(log(cells( :)),log(error( :)),); loglog(cells,error,legen{i});old on; switc data case ebar = ^(.3)*(./n).^(3); loglog(n,ebar,legen{3}) ais([^ *^ ^( 5) ^( )]) lgdstring = 'O(d3)'; case ebar = ^(.4)*(./n).^(); loglog(n,ebar,legen{3}) ais([^ *^ ^( 3) ^( )]) lgdstring = 'O(d)'; case 3 ebar = ^()*(./n).^(); loglog(n,ebar,legen{3}) ais([^ *^ ^( ) ^( )]) lgdstring = 'O(d)'; label('cells');ylabel('error');grid on; leg(types{},types{},lgdstring); slope(:,) name = ['data_',nmstr(data),'k',nmstr(k),'_acc.tikz']; matlabtikz(name, 'eigt', '\figreeigt', 'widt', '\figrewidt'); fnction [ k,x,nx,t,nt ] = Setp() k = *; X = ::; nx = nmel(x); T = :k:; nt = nmel(t) ; fnction Q = InitialData(data,X,nX,d) % Initial pressre distribtion switc data case H HU zeros(size()); case H HU *ones(size()); case 3 H (<).*.5 + (>=).* ; HU zeros(size()); % Integrate data Q = zeros(,nx); for i = :nmel(x) Q(,i) = integral(h,x(i) d,x(i)+d,'abstol',e )/(d*); Q(,i) = integral(hu,x(i) d,x(i)+d,'abstol',e )/(d*); 7

8 fnction [ Q,X ] = GetRef(data,g,K) Name = ['Q_',nmstr(data),'_k',nmstr(K),'.mat'] if eist(name, 'file') == ; load(name); else K = ; C = GetC(K); =.; wstring = 'Compting erence soltion'; w = waitbar(,wstring); [k,x,nx,t,nt] = Setp(); Q = initialdata(data,x,nx,/); F = zeros(,nx+); for t = :nt Q = SSPRK3(F,C,,nX,X,Q,K,g,k,'Roes Metod'); waitbar(t/nt); close(w); save(name,'q','x'); fnction [ Q ] = SSPRK3(F,C,,nX,X,Q,K,g,k,type) % Do first step in te Rnge Ktta Qedge(,:) = WENO(C,,nX,X,Q(,:),K); Qedge(,:) = WENO(C,,nX,X,Q(,:),K); F = Fl(F,Qedge,nX,g,,k,type); Y = Q(:,:) k/*(f(:,:nx+) F(:,:nX)); % Do second step in te Rnge Ktta Yedge(,:) = WENO(C,,nX,X,Y(,:),K); Yedge(,:) = WENO(C,,nX,X,Y(,:),K); F = Fl(F,Yedge,nX,g,,k,type); Y = (3/4)*Q(:,:)+.5*Y.5*k/*(F(:,:nX+) F(:,:nX)); % Do te tird step in te Rnge Ktta Yedge(,:) = WENO(C,,nX,X,Y(,:),K); Yedge(,:) = WENO(C,,nX,X,Y(,:),K); F = Fl(F,Yedge,nX,g,,k,type); Q = (/3)*Q(:,:)+(/3)*Y (/3)*k/*(F(:,:nX+) F(:,:nX)); fnction F = Fl(F,Qedge,nX,g,,k,type) switc type case 'La Friedrics' for i = :nx+ a = Qedge(:,*i ); b = Qedge(:,*i); F(:,i) = *(/k)*(a b)+*(f(a,g)+f(b,g)); case 'Roes Metod' for i = :nx+ Ql = Qedge(:,*i ); Qr = Qedge(:,*i); rql = sqrt(ql); rqr = sqrt(qr); avg = (Ql()/rQl()+Qr()/rQr())/(rQl()+rQr()); cavg = sqrt(g*(ql()+qr())/); L = [avg cavg,;,avg+cavg]; A = [,;L(,),L(,)] * abs(l) * /cavg*[l(,), ; L(,),]; F(:,i) = *(f(ql,g)+f(qr,g)) *A*(Qr Ql); 8

9 fnction f = f(q,g) f = [Q(,:);Q(,:).^./Q(,:)+*g*Q(,:).^]; fnction [Uedge,Xedge] = WENO(C,,nX,X,,K) % Add gost points, modify tis line according to initial condition type Ug = [(nx K:nX ),,(:K+)]; % Calclate te vales at te edges of eac cell for eac of te K stencils U = zeros(*nx+,k); for j = :K r = j ; U(,j) = sm( C(r+,:).* Ug((K r):(*k r)) ); for i = (K+):(nX+K) U((i K)*,j) = sm( C(r+,:).* Ug((i r):(i+k r)) ); U((i K)*+,j) = sm( C(r+,:).* Ug((i r):(i+k r)) ); U(*nX+,j) = sm( C(r+,:).* Ug((nX+K+ r):(nx+k++k r)) ); % Find weigts Wedge for Uedge W = FindWeigts(nX,Ug,K); % Apply weigts and retrn Uedge Uedge = zeros(*nx+,); for i = :(*nx+) Uedge(i) = sm(u(i,:).*w(i,:)); % Pass tis, somebody migt want it.. Xedge = resape([x /,X()+/;X /,X()+/],*nX+,); fnction [ Wedge ] = GetW(K,A,B,d,e) Wedge = zeros(,k); for j = :K A(j) = d(j)/((e+b(j))^); for j = :K Wedge(,j) = A(j)/sm(A); fnction [ B ] = GetB(B,U,K) % Retrn Beta coefficients switc K case B() = (U(3) U())^; B() = (U() U())^; case 3 B() = (3/)*(U(3) *U(4)+U(5))^ +.5*(3*U(3) 4*U(4)+U(5))^; B() = (3/)*(U() *U(3)+U(4))^ +.5*(U() U(4))^; B(3) = (3/)*(U() *U()+U(3))^ +.5*(U() 4*U()+3*U(3))^; oterwise error('only WENO coefficients for K = {,3} available'); fnction [ C ] = GetC(K) % Calclate matri wit interpolation coefficients C = zeros(k+,k); for r = ::K for j = :K 9

10 temp = ; for m = j+:k % Sm denominator C = ; for l = :K if l ~= m temp = ; for q = :K if (q ~= m) && (q ~= l) temp = temp*(r q+); C = C + temp; % Sm nmerator temp = ; for l = :K if l ~= m temp = temp*(m l); C = temp; % Devide and add temp = temp + C/C; C(r+,j+) = temp; fnction [ d ] = GetD(K) % Retrn d coefficients d = zeros(,k); switc K case d() = /3; d() = /3; d() = 3/4; d() = /4; case 3 d() = 3/; d() = 3/5; d(3) = /; oterwise error('only WENO coefficients for K = {,3} available');