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[WIP] Energy preserving #67

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6ab5752
Energy-preserving non linear conditions
Sondar74 Sep 19, 2023
fbd9bdd
Nonlinear constraints
Sondar74 Sep 24, 2023
c1fe866
Conditions for PEP 7
Sondar74 Nov 21, 2023
53a9a97
Eliminate duplication
Sondar74 Nov 22, 2023
7a8eaa1
Fix PEP 7th order
Sondar74 Nov 23, 2023
48eebed
Nonlinear constraints PEP order 7
Sondar74 Dec 5, 2023
7b06e40
PEP7
Sondar74 Dec 25, 2023
e252998
instructions
Sondar74 Dec 25, 2023
b2f1fa3
restore
Sondar74 Dec 25, 2023
e28e460
restore
Sondar74 Dec 25, 2023
ab74270
Generate q=7 constraints
Sondar74 Jan 16, 2024
bb658bf
Project and Manifest
Sondar74 Mar 14, 2024
0f84f6f
Update RK-coeff-opt/PEP_Generator/README.txt
Sondar74 Mar 14, 2024
8e48274
closing PR
Sondar74 Apr 15, 2024
304fd03
Merge branch 'EnergyPreserving' of github.com:Sondar74/RK-Opt into En…
Sondar74 Apr 15, 2024
7c2bba3
Update RK-coeff-opt/nonlinear_constraints.m
Sondar74 Apr 16, 2024
e57a2c5
Error message
Sondar74 Apr 18, 2024
c1fd47c
Move message to the end
Sondar74 Apr 19, 2024
ec04e9c
Update conditions 6 and 7
Sondar74 May 7, 2024
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65 changes: 65 additions & 0 deletions RK-coeff-opt/nonlinear_constraints.m
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Original file line number Diff line number Diff line change
Expand Up @@ -120,3 +120,68 @@
res = polyval(poly_coef, constrain_emb_stability);
con = [con, res .* conj(res) - 1];
end

q=1;

% PEP conditions

if q>=1
coneq(end+1) = sum(b)-1;
end
if q>=2
coneq(end+1) = c'*b-1/2;
end
if q >= 3
coneq(end+1) = c'.^2*b-1/3;
end
if q >= 4
coneq(end+1) = b'*A^2*c-b'*A*c+1/8;
coneq(end+1) = c'.^3*b-1/4;
coneq(end+1) = ((b'.*c')*A*c)-(b'*A*c.^2)/2-1/12;
end
if q >= 5
coneq(end+1) = (b'*A^2*c.^2)/2 + (b.*c)'*A^2*c-b'*A^2*c-(b'*A*c.^2)/2 + (b'*A*c)/2 - 1/24;
coneq(end+1) = c'.^4*b-1/5;
coneq(end+1) = 2*(b'*A*diag(c)*A*c)-(b'*(A*c).^2) - (b'*A^2*c)-(b'*A*c.^2)+(b'*A*c)-1/24;
coneq(end+1) = (b.*c.^2)'*A*c-(b'*A*c.^3)/3 - 1/12;
end
if q >= 6
coneq(end+1) = -(b'*A*c)^2 + (b'*A^2*c)- 2 *(b'*A^3*c)+ 2* (b'*A^4*c);
coneq(end+1) = -(b'*diag(c)*A^3*c) + (b'*A^3*c.^2)/2 + (b'*A^2*c)/3 + ((b.*c)'*A^2*c)/2 - (b'*A^2*c.^2)/4 - (b'*A*c)/4 - ((b'.*c')*A*c)/12 + (b'*A*c.^2)/24 + 1/36;
coneq(end+1) = -(b'*A*diag(c)*A^2*c) + (b'*A^2*diag(c)*A*c) + (b'*((A^2*c).*(A*c))) + (b'*A^2*c)/6 - (b'*A^2*c.^2)/2 - (b'*(A*c).^2)/2 - (b'*A*c)/6 + (b'*A*c.^2)/4 + 1/48;
coneq(end+1) = (b'*A^2*c)/6 - ((b.*c)'*A^2*c)/2 + (b'*diag(c).^2*A^2*c)/2 - (b'*A^2*c.^2)/4 + (b'*A^2*c.^3)/6 - (b'*A*c)/12 + ((b'.*c')*A*c)/6 - ((b.*c.^2)'*A*c)/4 + (b'*A*c.^2)/6 - (b'*A*c.^3)/12 + 1/144;
coneq(end+1) = (b'*A^2*c)/3 - ((b.*c)'*A^2*c)/2 - (b'*A*diag(c)*A*c) + (b'*diag(c)*A*diag(c)*A*c) - (b'*A^2*c.^2)/4 + (b'*A*diag(c)*A*c.^2)/2 + (b'*(A*c).^2)/2 - (b'*A*c)/12 + ((b'.*c')*A*c)/4 - (b'*((A*c.^2).*(A*c)))/2 + 3*(b'*A*c.^2)/8 - ((b.*c)'*A*c.^2)/2 - 1/48;
coneq(end+1) = (b'*A^2*c)/6 - ((b.*c)'*A^2*c)/2 - (b'*A^2*c.^2)/4 + (b'*diag(c)*A^2*c.^2)/2 + (b'*A*c)/12 + ((b'.*c')*A*c)/12 - (b'*A*c.^2)/24 - 1/72;
coneq(end+1) = (b'*A*(A*c).^2)/2 + (b'*A^2*c)/12 - (b'*A*diag(c)*A*c)/2 - (b'*(A*c).^2)/4 - (b'*A*c)/12 + ((b'.*c')*A*c)/3 + (b'*A*c.^2)/12 - 5/288;
coneq(end+1) = (b'*A^2*c)/12 - (b'*A*diag(c)*A*c)/2 + (b'*A*diag(c).^2*A*c)/2 + (b'*(A*c).^2)/4 - (b'*diag(c)*(A*c).^2)/2 - (b'*A*c)/12 + ((b'.*c')*A*c)/12 + ((b.*c.^2)'*A*c)/4 + 5*(b'*A*c.^2)/24 - (b'*A*c.^3)/4 - 5/288;
coneq(end+1) = -((b'.*c')*A*c)/12 + ((b.*c.^2)'*A*c)/4 - (b'*diag(c).^3*A*c)/6 + (b'*A*c.^2)/24 - (b'*A*c.^3)/12 + (b'*A*c.^4)/24 - 1/720;
coneq(end+1) = -((b'.*c')*A*c)/12 + ((b.*c.^2)'*A*c)/4 + (b'*A*c.^2)/24 - (b'*diag(c).^2*A*c.^2)/4 - (b'*A*c.^3)/12 + (b'*diag(c)*A*c.^3)/6 - 1/144;
coneq(end+1) = (c'.^5*b)/120 - 1/720;
end
if q >= 7
coneq(end+1) = -(b'*A^4*c) + (b'*((A*(A*(A*(A*c)))).*c)) + (b'*((A*(A*(A*(A*c.^2))))))/2 + (b'*A^3*c)/2 - (b'*diag(c)*A^3*c)/2 - (b'*A^3*c.^2)/4 - (b'*A^2*c)/12 + ((b.*c)'*A^2*c)/12 + (b'*A^2*c.^2)/24 - ((b'*A*c)/6 - 1/36)*((b'*A*c) - 1/6) - ((b'*A*c)/3 - 1/18)*((b'*A*c) - 1/6) + ((b'*A*c)/2 - 1/12)*((b'*A*c) - 1/6) - ((b'*A*c)/2 - 1/12)*(-(b'*A*c)/2 + ((b'.*c')*A*c) - 1/24) - ((b'*A*c) - 1/6)*(-(b'*A*c)/2 + (b'*A*c.^2)/2 + 1/24)/2;
coneq(end+1) = -(b'*A^4*c) + (b'*((A*(c.*(A*(A*(A*c))))))) + (b'*((A*(A*(A*(c.*(A*c))))))) - (b'*((A*(A*(A*c))).*(A*c))) + (b'*A^3*c) - (b'*A*diag(c)*A^2*c)/2 - (b'*A^2*diag(c)*A*c)/2 - (b'*A^3*c.^2)/2 + (b'*((A^2*c).*(A*c)))/2 - (b'*A^2*c)/2 + ((b.*c)'*A^2*c)/6 + (b'*A*diag(c)*A*c)/6 + (b'*A^2*c.^2)/3 - (b'*(A*c).^2)/12 + (b'*A*c)/8 - ((b'.*c')*A*c)/12 - (b'*A*c.^2)/12 - ((b'*A*c)/6 - 1/36)*((b'*A*c) - 1/6) - ((b'*A*c)/3 - 1/18)*((b'*A*c) - 1/6) + ((b'*A*c)/2 - 1/12)*((b'*A*c) - 1/6) - ((b'*A*c)/2 - 1/12)*(-(b'*A*c) + (b'*A*c.^2) + 1/12) + ((b'*A*c)/2 - 1/12)*(-(b'*A*c)/2 + ((b'.*c')*A*c) - 1/24) - ((b'*A*c) - 1/6)*(-(b'*A*c)/4 + ((b'.*c')*A*c)/2 - 1/48);
coneq(end+1) = -(b'*A^4*c)/2 + (b'*((A*(A*diag(c)*(A*(A*c)))))) + (b'*A^3*c)/2 - (b'*A*diag(c)*A^2*c)/2 - (b'*A^2*diag(c)*A*c)/2 + (b'*((A*(A*c)).*(A*(A*c))))/2 - (b'*((A^2*c).*(A*c)))/2 - (b'*A^2*c)/4 + ((b.*c)'*A^2*c)/6 + (b'*A*diag(c)*A*c)/3 + (b'*A^2*c.^2)/12 + (b'*(A*c).^2)/12 + (b'*A*c)/12 - ((b'.*c')*A*c)/12 - (b'*A*c.^2)/12 - ((b'*A*c)/6 - 1/36)*((b'*A*c) - 1/6) - 5*((b'*A*c)/3 - 1/18)*((b'*A*c) - 1/6)/2 + 3*((b'*A*c)/2 - 1/12)*((b'*A*c) - 1/6)/2 - 2*((b'*A*c)/2 - 1/12)*((b'*A^2*c) - (b'*A*c) + 1/8);
coneq(end+1) = (b'*diag(c)*A^3*c)/2 - (b'*((A*(A*(A*c))).*c.^2))/2 - (b'*A^3*c.^2)/4 + (b'*((A*(A*(A*c.^3)))))/6 - (b'*A^2*c)/12 - ((b.*c)'*A^2*c)/6 + (b'*diag(c).^2*A^2*c)/4 + (b'*A^2*c.^2)/6 - (b'*A^2*c.^3)/12 + (b'*A*c)/24 - ((b.*c.^2)'*A*c)/24 - (b'*A*c.^2)/24 + (b'*A*c.^3)/72;
coneq(end+1) = (b'*diag(c)*A^3*c)/2 + (b'*A*diag(c)*A^2*c)/2 - (b'*((A*(c.*(A*(A*c)))).*c)) - (b'*A^2*diag(c)*A*c)/2 - (b'*A^3*c.^2)/4 + (b'*((A*(A*(c.*(A*c.^2))))))/2 - (b'*((A^2*c).*(A*c)))/2 + (b'*((A*(A*c)).*(A*c.^2)))/2 - (b'*A^2*c)/4 + (b'*A*diag(c)*A*c)/6 + (b'*diag(c)*A*diag(c)*A*c)/2 + (b'*A^2*c.^2)/4 - (b'*A*diag(c)*A*c.^2)/4 + (b'*(A*c).^2)/6 + (b'*A*c)/6 - ((b'.*c')*A*c)/6 - (b'*((A*c.^2).*(A*c)))/4 - (b'*A*c.^2)/8 - ((b'*A*c)/2 - 1/12)*(-(b'*A*c) + (b'*A*c.^2) + 1/12)/2 + ((b'*A*c)/2 - 1/12)*(-(b'*A*c)/2 + ((b'.*c')*A*c) - 1/24);
coneq(end+1) = (b'*diag(c)*A^3*c)/2 - (b'*A*diag(c)*A^2*c)/2 + (b'*A^2*diag(c)*A*c)/2 - (b'*((A*(A*(c.*(A*c)))).*c)) - (b'*A^3*c.^2)/4 + (b'*((A*(c.*(A*(A*c.^2))))))/2 + (b'*((A^2*c).*(A*c)))/2 - (b'*A^2*c)/4 - ((b.*c)'*A^2*c)/12 + (b'*A*diag(c)*A*c)/6 - (b'*((A*(A*c.^2)).*(A*c)))/2 - (b'*A^2*c.^2)/24 + (b'*diag(c)*A^2*c.^2)/2 - (b'*(A*c).^2)/12 + (b'*A*c)/8 - ((b'.*c')*A*c)/12 - (b'*A*c.^2)/12;
coneq(end+1) = b'*((A*(A*((A*c).*(A*c)))))/2 - (b'*((A*((A*c).*(A*(A*c)))))) + (b'*A*diag(c)*A^2*c)/2 - (b'*A^2*diag(c)*A*c)/2 + (b'*A*(A*c).^2)/4 + (b'*((A^2*c).*(A*c)))/2 - ((b.*c)'*A^2*c)/3 + (b'*A*diag(c)*A*c)/12 + (b'*A^2*c.^2)/12 - 7*(b'*(A*c).^2)/24 + (b'*A*c)/24 + ((b'.*c')*A*c)/6 - (b'*A*c.^2)/12 + ((b'*A*c)/6 - 1/36)*((b'*A*c) - 1/6) + ((b'*A*c)/3 - 1/18)*((b'*A*c) - 1/6) - ((b'*A*c)/2 - 1/12)*((b'*A*c) - 1/6) + ((b'*A*c)/2 - 1/12)*((b'*A^2*c) - (b'*A*c) + 1/8) - 1/144;
coneq(end+1) = (b'*A*diag(c)*A^2*c)/2 - (b'*((A*(c.^2.*(A*(A*c))))))/2 - (b'*A^2*diag(c)*A*c)/2 + (b'*((A*(A*(c.^2.*(A*c))))))/2 - (b'*((A^2*c).*(A*c)))/2 + (b'*((A*(A*c)).*(A*c).*c)) - ((b.*c)'*A^2*c)/12 - (b'*diag(c).^2*A^2*c)/4 + (b'*A*diag(c)*A*c)/6 + 5*(b'*A^2*c.^2)/24 - (b'*A^2*c.^3)/4 + (b'*(A*c).^2)/6 - (b'*diag(c)*(A*c).^2)/2 + (b'*A*c)/24 + ((b'.*c')*A*c)/12 + ((b.*c.^2)'*A*c)/8 - 5*(b'*A*c.^2)/24 + (b'*A*c.^3)/8 + ((b'*A*c)/2 - 1/12)*(-(b'*A*c) + (b'*A*c.^2) + 1/12)/2 - ((b'*A*c)/2 - 1/12)*(-(b'*A*c)/2 + ((b'.*c')*A*c) - 1/24) - 1/144;
coneq(end+1) = (b'*A^3*c)/6 - (b'*A*diag(c)*A^2*c)/2 - (b'*A^2*diag(c)*A*c)/2 + (b'*((A*diag(c)*(A*diag(c)*(A*c))))) - (b'*((A^2*c).*(A*c)))/2 - (b'*A^2*c)/6 + ((b.*c)'*A^2*c)/2 + (b'*((A*diag(c)*(A*c)).*(A*c))) + 2*(b'*A*diag(c)*A*c)/3 - (b'*diag(c)*A*diag(c)*A*c) + (b'*A^2*c.^2)/6 - (b'*A*diag(c)*A*c.^2)/2 + (b'*(A*c).^2)/4 - (b'*A*c)/180 - ((b'.*c')*A*c)/4 - (b'*((A*c.^2).*(A*c)))/2 - (b'*A*c.^2)/6 + ((b.*c)'*A*c.^2)/2 - 2*((b'*A*c)/6 - 1/36)*((b'*A*c) - 1/6) + ((b'*A*c)/2 - 1/12)*((b'*A*c) - 1/6) - 2*((b'*A*c) - 1/6)*(-(b'*A*c)/4 + ((b'.*c')*A*c)/2 - 1/48) + 149/15120;
coneq(end+1) = ((b.*c)'*A^2*c)/12 - (b'*diag(c).^2*A^2*c)/4 + (b'*((A*(A*c)).*c.^3))/6 + (b'*A^2*c.^2)/24 - (b'*A^2*c.^3)/12 + (b'*((A*(A*c.^4))))/24 + ((b.*c.^2)'*A*c)/12 - (b'*diag(c).^3*A*c)/12 - (b'*A*c.^2)/24 + (b'*A*c.^3)/18 - (b'*A*c.^4)/48;
coneq(end+1) = ((b.*c)'*A^2*c)/6 - (b'*diag(c).^2*A^2*c)/4 + (b'*A*diag(c)*A*c)/6 - (b'*diag(c)*A*diag(c)*A*c)/2 + (b'*((A*(c.*(A*c))).*c.^2))/2 + (b'*A^2*c.^2)/12 - (b'*A*diag(c)*A*c.^2)/4 - (b'*A^2*c.^3)/12 + (b'*((A*(c.*(A*c.^3)))))/6 - (b'*(A*c).^2)/12 - ((b'.*c')*A*c)/12 + ((b.*c.^2)'*A*c)/8 + (b'*((A*c.^2).*(A*c)))/4 - (b'*A*c.^2)/12 + ((b.*c)'*A*c.^2)/4 - (b'*diag(c).^2*A*c.^2)/4 - (b'*((A*c.^3).*(A*c)))/6 + (b'*A*c.^3)/24;
coneq(end+1) = ((b.*c)'*A^2*c)/4 - (b'*diag(c).^2*A^2*c)/4 + (b'*A^2*c.^2)/8 - (b'*diag(c)*A^2*c.^2)/2 + (b'*((A*(A*c.^2)).*c.^2))/4 - (b'*A^2*c.^3)/12 + (b'*((A*(A*c.^3)).*c))/6 - (b'*A*c)/24 + ((b.*c.^2)'*A*c)/24 + (b'*A*c.^2)/24 - (b'*A*c.^3)/72;
coneq(end+1) = -3*(b'*A*(A*c).^2)/4 + (b'*((A*((A*c).*(A*c))).*c))/2 + ((b.*c)'*A^2*c)/12 + 5*(b'*A*diag(c)*A*c)/12 - (b'*diag(c)*A*diag(c)*A*c)/2 + (b'*((A*(c.^2.*(A*(A*c))))))/2 + (b'*A^2*c.^2)/24 - (b'*A*diag(c)*A*c.^2)/4 + 7*(b'*(A*c).^2)/24 - (b'*A*c)/24 - ((b'.*c')*A*c)/6 - (b'*((A*c.^2).*(A*c)))/4 - (b'*A*c.^2)/24 + ((b.*c)'*A*c.^2)/4 + 1/144;
coneq(end+1) = ((b.*c)'*A^2*c)/12 + (b'*A*diag(c)*A*c)/3 - (b'*diag(c)*A*diag(c)*A*c)/2 - (b'*A*diag(c).^2*A*c)/2 + (b'*((A*(c.^2.*(A*c))).*c))/2 + (b'*A^2*c.^2)/24 - (b'*A*diag(c)*A*c.^2)/4 + (b'*((A*(c.^2.*(A*c.^2)))))/4 - (b'*(A*c).^2)/6 + (b'*diag(c)*(A*c).^2)/2 - (b'*A*c)/24 - ((b'.*c')*A*c)/12 - ((b.*c.^2)'*A*c)/8 + (b'*((A*c.^2).*(A*c)))/4 - (b'*((A*c.^2).*(A*c).*c))/2 - (b'*A*c.^2)/24 + ((b.*c)'*A*c.^2)/4 + (b'*diag(c).^2*A*c.^2)/8 + (b'*A*c.^3)/8 - (b'*diag(c)*A*c.^3)/4 + 1/144;
coneq(end+1) = ((b.*c)'*A^2*c)/6 + (b'*A*diag(c)*A*c)/6 - (b'*diag(c)*A*diag(c)*A*c)/2 + (b'*A^2*c.^2)/12 - (b'*diag(c)*A^2*c.^2)/4 - (b'*A*diag(c)*A*c.^2)/4 + (b'*((A*(c.*(A*c.^2))).*c))/2 - (b'*(A*c).^2)/12 - (b'*A*c)/12 + ((b'.*c')*A*c)/12 + (b'*((A*c.^2).*(A*c)))/4 - (b'*((A*c.^2).*(A*c.^2)))/8 + (b'*A*c.^2)/24;
coneq(end+1) = -(b'*A*(A*c).^2)/4 + (b'*((A*(c.*(A*c).*(A*c)))))/2 + (b'*A*diag(c)*A*c)/4 - (b'*A*diag(c).^2*A*c)/2 - (b'*((A*c).*(A*c).*(A*c)))/6 + (b'*(A*c).^2)/8 - ((b'.*c')*A*c)/4 + ((b.*c.^2)'*A*c)/4 + (b'*A*c.^3)/12;
coneq(end+1) = (b'*A*diag(c)*A*c)/12 - (b'*A*diag(c).^2*A*c)/4 + (b'*((A*(c.^3.*(A*c)))))/6 - (b'*(A*c).^2)/24 + (b'*diag(c)*(A*c).^2)/4 - (b'*((A*c).*(A*c).*c.^2))/4 - ((b'.*c')*A*c)/12 - ((b.*c.^2)'*A*c)/24 + (b'*diag(c).^3*A*c)/6 + 7*(b'*A*c.^3)/72 - (b'*A*c.^4)/12 - 1/1440;
coneq(end+1) = -((b.*c.^2)'*A*c)/24 + (b'*diag(c).^3*A*c)/12 - (b'*((A*c).*c.^4))/24 + (b'*A*c.^3)/72 - (b'*A*c.^4)/48 + (b'*((A*c.^5)))/120;
coneq(end+1) = -((b.*c.^2)'*A*c)/12 + (b'*diag(c).^3*A*c)/12 + (b'*diag(c).^2*A*c.^2)/8 - (b'*((A*c.^2).*c.^3))/12 + (b'*A*c.^3)/36 - (b'*diag(c)*A*c.^3)/12 - (b'*A*c.^4)/48 + (b'*((A*c.^4).*c))/24;
coneq(end+1) = b'*(c.^6)/720 - 1/5040;
end
% Throw error message in case q>7
msg = 'The energy-preserving conditions for orders q>7 have not been implemented.';
if q >= 8
error(msg);
end