6、 Explicit Numerical Solver of the Heat Equation
6.3 Mass conservation
We developed an interest in doing justice to the variation of total mass in our numerical solver due to
∂Γ
∂t = ∆sΓ = ∇s2Γ Integrate both sides,
∂Γ
∂tdV
Ω
= ∇s2ΓdV
Ω
(46)
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From [16],
= ∇2Γ−∂2Γ
∂n2−∂Γ
∂ndV
Ω
We impose our assumption
∂Γ
∂n= 0 It will lead to
∂2Γ
∂n2 = 0 So, (46) becomes
∂Γ
∂tdV
Ω
= ∇2ΓdV
Ω
= ∇ ∙ ∇Γ dV
Ω
According to the Divergence Theorem as follows,
∇ ∙ Γd𝐗
Ω
= Γ ∙ nd𝐀
∂Ω
So, (46) becomes
∂Γ
∂tdV
Ω
= ∇Γ ∙ ndS
∂Ω
= n ∙ ∇ΓdS
∂Ω
= ∂Γ
∂ndS
∂Ω
= 0 Here, we have the result
∂Γ
∂tdV
Ω
= 0
Transform coordinate in integration and V = S, Ω = ∂Ω in our domain
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After discretizing, it turns to be
ui,j ∂𝐗i,j like Section 3 and Section 4 which compute till T = 5. Because it will cost us too much time.
6.3.1 The Mass of the Solvers with Section 6.1 and Central Difference
Method 1
Use the solver of the heat equation (45) Γt = ΔsΓ on the spherical surface domain by Section 6.1 with central difference method 1. Initial settings: M = 32, N = 64, R = 5,
∆t =12 1
2π 2∆x2 =12N12. Domain is the same as Figure 7.
Case 1: Initial setting is the same as Figure 8,
Figure 88: Left, T = 0.25, mass = 14.7325, relative error = 2.4338e − 005.
Mid-left, T = 0.5, mass = 14.7325, relative error = 2.4076e − 005.
Mid-right, T = 0.75, mass = 14.7325, relative error = 2.3819e − 005.
Right, T = 0.1, mass = 14.7325, relative error = 2.3562e − 005.
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Figure 89: T = 5, mass = 14.7324, relative error = 1.8856e − 005.
Figure 90: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 88, Figure 89, Figure 90, u was diffused as time goes by to be like the look of the original domain. Total mass is getting decreasing a little bit, but it will turn to stable that means mass will not change any more in a long time. Case 1 does not comply with the mass conservation law.
Case 2: Initial setting is the same as Figure 12,
Figure 91: Left, T = 1/64, mass = 0.24067, relative error = 4.2098e − 006.
Mid-left, T = 1/32, mass = 0.24067, relative error = 7.2202e − 006.
Mid-right, T = 3/64, mass = 0.24067, relative error = 1.0231e − 005.
Right, T = 1/16, mass = 0.24067, relative error = 1.3241e − 005.
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Figure 92: T = 5, mass = 0.24097, relative error = 0.0012786.
Figure 93: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 91, Figure 92, Figure 93, u was diffused as time goes by to be like the look of the original domain. Total mass is getting increasing a little bit, but it will turn to stable that means mass will not change any more in a long time. Case 2 does not comply with the mass conservation law.
Case 3: Initial setting is the same as Figure 16,
Figure 94: Left, T = 0.25, mass = 6.7316, relative error = 0.0051562.
Mid-left, T = 0.5, mass = 6.6839, relative error = 0.012208.
Mid-right, T = 0.75, mass = 6.6464, relative error = 0.017755.
Right, T = 1, mass = 6.6195, relative error = 0.021727.
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Figure 95: T = 5, mass = 6.526, relative error = 0.035539.
Figure 96: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 94, Figure 95, Figure 96, u was diffused as time goes by to be like the look of the original domain. Total mass is getting decreasing, but it will turn to stable that means mass will not change any more in a long time. Case 3 does not comply with the mass conservation law.
Case 4: Initial setting is the same as Figure 20,
Figure 97: Left, T = 0.25, mass = 446.7228, relative error = 0.00014121.
Mid-left, T = 0.5, mass = 446.7606, relative error = 0.0002258.
Mid-right, T = 0.75, mass = 446.7887, relative error = 0.00028867.
Right, T = 1, mass = 446.8119, relative error = 0.00034063.
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Figure 98: T = 5, mass = 446.9892, relative error = 0.00073756.
Figure 99: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 97, Figure 98, Figure 99, u was diffused as time goes by to be like the look of the original domain. Total mass is getting increasing, but it will turn to stable that means mass will not change any more in a long time. Case 4 does not comply with the mass conservation law.
In above Case 1~Case 4, we derive that Case 1 can get the better result by mass conservation with some machine error；the others show up a little bit derivation in our relative error.
6.3.2 The Mass of the Solver with Section 6.1 and Central Difference
Method 2
Use the solver of the heat equation (45) Γt = ΔsΓ on the spherical surface domain by Section 6.1 with central difference method 2. Initial settings：M = 32, N = 64, R = 5,
∆t =12 1
2π 2∆x2 =12N12. Domain is the same as Figure 7.
Case 1: Initial setting is the same as Figure 8,
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Figure 100: Left, T = 0.25, mass = 14.7325, relative error = 2.428e − 005.
Mid-left, T = 0.5, mass = 14.7325, relative error = 2.4019e − 005.
Mid-right, T = 0.75, mass = 14.7325, relative error = 2.3764e − 005.
Right, T = 0.1, mass = 14.7325, relative error = 2.3508e − 005.
Figure 101: T = 5, mass = 14.7324, relative error = 1.8826e − 005.
Figure 102: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 100, Figure 101, Figure 102, u was diffused as time goes by to be like the look of the original domain. Total mass is getting decreasing a little bit, but it will turn to stable that means mass will not change any more in a long time. Case 1 does not comply with the mass conservation law.
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Case 2: Initial setting is the same as Figure 12,
Figure 103: Left, T = 1/64, mass = 0.24067, relative error = 4.1997e − 006.
Mid-left, T = 1/32, mass = 0.24067, relative error = 7.2028e − 006.
Mid-right, T = 3/64, mass = 0.24067, relative error = 1.0206e − 005.
Right, T = 1/16, mass = 0.24067, relative error = 1.3209e − 005.
Figure 104: T = 5, mass = 0.24097, relative error = 0.001275.
Figure 105: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 103, Figure 104, Figure 105, u was diffused as time goes by to be like the look of the original domain. Total mass is getting increasing a little bit, but it will turn to stable that means mass will not change any more in a long time. Case 2 does not comply with the mass conservation law.
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Case 3: Initial setting is the same as Figure 16,
Figure 106: Left, T = 0.25, mass = 6.7317, relative error = 0.005139.
Mid-left, T = 0.5, mass = 6.6841, relative error = 0.012178.
Mid-right, T = 0.75, mass = 6.6466, relative error = 0.017721.
Right, T = 1, mass = 6.6197, relative error = 0.021695.
Figure 107: T = 5, mass = 6.5261, relative error = 0.035528.
Figure 108: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 106, Figure 107, Figure 108, u was diffused as time goes by to be like the look of the original domain. Total mass is getting decreasing, but it will turn to stable that means mass will not change any more in a long time. Case 3 does not comply with the mass conservation law.
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Case 4: Initial setting is the same as Figure 20,
Figure 109: Left, T = 0.25, mass = 446.7227, relative error = 0.00014096.
Mid-left, T = 0.5, mass = 446.7604, relative error = 0.00022546.
Mid-right, T = 0.75, mass = 446.7885, relative error = 0.00028826.
Right, T = 1, mass = 446.8117, relative error = 0.00034017.
Figure 110: T = 5, mass = 446.9889, relative error = 0.00073693.
Figure 111: Left, the values of u on the equator change with time.
Right, total mass of u changes with time in 0~5 seconds.
According to Figure 109, Figure 110, Figure 111, u was diffused as time goes by to be like the look of the original domain. Total mass is getting increasing, but it will turn to stable that means mass will not change any more in a long time. Case 4 does not comply with the mass conservation law.
In above Case 1~Case 4, we derive that Case 1 can get the better result by mass conservation
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with some machine error；the others show up a little bit derivation that is less than one percent in our relative error.