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<Text-field style="Heading 1" layout="Heading 1"><Font executable="true">Constraint transport scheme in GRHydro</Font></Text-field>
<Text-field style="Text" layout="Heading 1">The current constraint transport scheme in GRHydro is based on WhiskyMHD (arXiv:gr-qc/0701109) and Bruno Giacomazzo's PhD thesis (http://www.sissa.it/ap/PhD/Theses/giacomazzo.ps.gz). It assumes that all magnetic field components live at the cell center and uses the Reconstruction step to move them to the cell interfaces.</Text-field>
<Text-field style="Text" layout="Heading 1"><Font encoding="UTF-8">This worksheet helps checking the expressions entered into the code by explicitely checking that div(\134dot B) = 0.</Font></Text-field>
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<Text-field prompt="> " style="Maple Input" layout="Normal">restart;</Text-field>
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<Text-field prompt="> " style="Maple Input" layout="Normal"># we later expand taylor series in eps which helps counting orders in the stepsize
dx:=eps*dX;dy:=eps*dY;dz:=eps*dZ;</Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Fluxes of the magnetic field components</Text-field></Title>
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<Text-field prompt="> " style="Maple Input" layout="Normal">BconsxfluxX := (i,j,k) -> 0;</Text-field>
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<Text-field prompt="> " style="Maple Input" layout="Normal">BconsyfluxX := (i,j,k) -> add(add(add(BconsyfluxX_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3);</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconszfluxX := (i,j,k) -> add(add(add(BconszfluxX_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3);</Text-field>
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<Text-field prompt="> " style="Maple Input" layout="Normal"># enforce F^I(B^j) = -F^j(B^i) if desired to check that RHS computed agrees to some order with the one used in divergence cleaning
BconsxfluxY := (i,j,k) -> add(add(add(BconsxfluxY_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3); #-BconsyfluxX(i,j,k);</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconsyfluxY := (i,j,k) -> 0;</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconszfluxY := (i,j,k) -> add(add(add(BconszfluxY_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3);</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconsxfluxZ := (i,j,k) -> add(add(add(BconsxfluxZ_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3); #-BconszfluxX(i,j,k);</Text-field>
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<Group labelreference="L12">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconsyfluxZ := (i,j,k) -> add(add(add(BconsyfluxZ_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3); #-BconszfluxY(i,j,k);</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">BconszfluxZ := (i,j,k) -> 0;</Text-field>
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<Text-field style="Heading 1" layout="Heading 1">Lapse</Text-field></Title>
<Group labelreference="L14">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">alp := (i,j,k) -> add(add(add(alp_coeff[_i,_j,_k] * (i*dx)^_i * (j*dy)^_j * (k*dz)^_k,_i=0..3),_j=0..3),_k=0..3);</Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Electric fields (eventually from E=v \134cross B=F(B))</Font></Text-field></Title>
<Group labelreference="L16">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Ey := (i,j,k) -> 1/4*(alp(i+1/2,j,k)*BconszfluxX(i+1/2,j,k) + alp(i+1/2,j,k+1)*BconszfluxX(i+1/2,j,k+1)) -
1/4*(alp(i,j,k+1/2)*BconsxfluxZ(i,j,k+1/2) + alp(i+1,j,k+1/2)*BconsxfluxZ(i+1,j,k+1/2));</Text-field>
</Input>
</Group>
<Group labelreference="L17">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Ez := (i,j,k) -> 1/4*(alp(i,j+1/2,k)*BconsxfluxY(i,j+1/2,k) + alp(i+1,j+1/2,k)*BconsxfluxY(i+1,j+1/2,k)) -
1/4*(alp(i+1/2,j,k)*BconsyfluxX(i+1/2,j,k) + alp(i+1/2,j+1,k)*BconsyfluxX(i+1/2,j+1,k));</Text-field>
</Input>
</Group>
<Group labelreference="L18">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Ex := (i,j,k) -> 1/4*(alp(i,j,k+1/2)*BconsyfluxZ(i,j,k+1/2) + alp(i,j+1,k+1/2)*BconsyfluxZ(i,j+1,k+1/2)) -
1/4*(alp(i,j+1/2,k)*BconszfluxY(i,j+1/2,k) + alp(i,j+1/2,k+1)*BconszfluxY(i,j+1/2,k+1));</Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">RHS for B</Text-field></Title>
<Group labelreference="L19">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Bconsrhsx := (i,j,k) -> -1/2*(Ey(i-1,j ,k-1)-Ey(i-1,j ,k ))/dz
-1/2*(Ez(i-1,j ,k )-Ez(i-1,j-1,k ))/dy
-1/2*(Ey(i ,j ,k-1)-Ey(i ,j ,k ))/dz
-1/2*(Ez(i ,j ,k )-Ez(i ,j-1,k ))/dy:</Text-field>
</Input>
</Group>
<Group labelreference="L20">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Bconsrhsy := (i,j,k) -> -1/2 * (Ez(i-1,j-1,k )-Ez(i ,j-1,k )) / dx
-1/2 * (Ex(i ,j-1,k )-Ex(i ,j-1,k-1)) / dz
-1/2 * (Ez(i-1,j ,k )-Ez(i ,j ,k )) / dx
-1/2 * (Ex(i ,j ,k )-Ex(i ,j ,k-1)) / dz:</Text-field>
</Input>
</Group>
<Group labelreference="L21">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Bconsrhsz := (i,j,k) -> - 1/2 * (Ex(i ,j-1,k-1)-Ex(i ,j ,k-1)) / dy
- 1/2 * (Ey(i ,j ,k-1)-Ey(i-1,j ,k-1)) / dx
- 1/2 * (Ex(i ,j-1,k )-Ex(i ,j ,k )) / dy
- 1/2 * (Ey(i ,j ,k )-Ey(i-1,j ,k )) / dx:</Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Check RHS at center of cell</Text-field></Title>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">Bconsrhsx(0,0,0):</Text-field>
</Input>
</Group>
<Group labelreference="L23">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">simplify(%):</Text-field>
</Input>
</Group>
<Group labelreference="L24">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">algsubs(eps=0,%);</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L40">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">(alp(0,-1/2,0)*BconsxfluxY(0,-1/2,0)-alp(0,1/2,0)*BconsxfluxY(0,1/2,0))/dy +
(alp(0,0,-1/2)*BconsxfluxZ(0,0,-1/2)-alp(0,0,1/2)*BconsxfluxZ(0,0,1/2))/dz:</Text-field>
</Input>
</Group>
<Group labelreference="L39">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">collect(%,eps):algsubs(eps^2=0,%);</Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Check constraint transport property for edge based divergence operator (see Bruno's thesis)</Text-field></Title>
<Group labelreference="L29">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">divB := (i,j,k) -> (Bconsrhsx(i+1,j,k)-Bconsrhsx(i,j,k)+Bconsrhsx(i+1,j+1,k)-Bconsrhsx(i,j+1,k)+Bconsrhsx(i+1,j,k+1)-Bconsrhsx(i,j,k+1)+Bconsrhsx(i+1,j+1,k+1)-Bconsrhsx(i,j+1,k+1))/4/dx +
(Bconsrhsy(i,j+1,k)-Bconsrhsy(i,j,k)+Bconsrhsy(i+1,j+1,k)-Bconsrhsy(i+1,j,k)+Bconsrhsy(i,j+1,k+1)-Bconsrhsy(i,j,k+1)+Bconsrhsy(i+1,j+1,k+1)-Bconsrhsy(i+1,j,k+1))/4/dy +
(Bconsrhsz(i,j,k+1)-Bconsrhsz(i,j,k)+Bconsrhsz(i+1,j,k+1)-Bconsrhsz(i+1,j,k)+Bconsrhsz(i,j+1,k+1)-Bconsrhsz(i,j+1,k)+Bconsrhsz(i+1,j+1,k+1)-Bconsrhsz(i+1,j+1,k))/4/dz;</Text-field>
</Input>
</Group>
<Group labelreference="L30">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">simplify(divB(0,0,0));</Text-field>
</Input>
</Group>
</Section>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Text-field superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="12" bold="false" subscript="false" family="Times New Roman" opaque="false" underline="false" background="[255,255,255]" readonly="false" foreground="[0,0,0]" alignment="left" firstindent="0" spacebelow="0" leftmargin="0" linespacing="0.0" initial="0" linebreak="space" rightmargin="0" bulletsuffix="" spaceabove="0" bullet="none" pagebreak-before="false"></Text-field>
<Text-field superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="12" bold="false" subscript="false" family="Times New Roman" opaque="false" underline="false" background="[255,255,255]" readonly="false" foreground="[0,0,0]" alignment="left" firstindent="0" spacebelow="0" leftmargin="0" linespacing="0.0" initial="0" linebreak="space" rightmargin="0" bulletsuffix="" spaceabove="0" bullet="none" pagebreak-before="false"></Text-field>
</Worksheet>
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