From a8f7c7db1545a0b4b007d4db3a88ed4757cb4264 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 24 Oct 2025 11:52:15 +0100 Subject: [PATCH 1/7] Rename demos --- demos/bbm_auxiliary/bbm_final.png | Bin 17682 -> 0 bytes demos/bbm_auxiliary/bbm_init.png | Bin 16767 -> 0 bytes demos/bbm_auxiliary/demo_bbm_aux.py.rst | 63 +++++++++--------- demos/bbm_auxiliary/invariant_errors.png | Bin 23449 -> 0 bytes demos/bbm_auxiliary/invariants.png | Bin 11156 -> 0 bytes docs/source/demos/bbm_aux_errors.png | Bin 0 -> 25219 bytes docs/source/demos/bbm_aux_final.png | Bin 0 -> 18952 bytes docs/source/demos/bbm_aux_init.png | Bin 0 -> 17964 bytes docs/source/demos/bbm_aux_invariants.png | Bin 0 -> 12360 bytes docs/source/demos/bbm_final.png | Bin 18973 -> 0 bytes docs/source/demos/bbm_hamiltonian_errors.png | Bin 0 -> 25191 bytes docs/source/demos/bbm_hamiltonian_final.png | Bin 0 -> 18893 bytes docs/source/demos/bbm_hamiltonian_init.png | Bin 0 -> 17988 bytes .../demos/bbm_hamiltonian_invariants.png | Bin 0 -> 12360 bytes docs/source/demos/bbm_init.png | Bin 17597 -> 0 bytes docs/source/demos/invariant_errors.png | Bin 23657 -> 0 bytes docs/source/demos/invariants.png | Bin 11364 -> 0 bytes docs/source/index.rst | 3 +- 18 files changed, 33 insertions(+), 33 deletions(-) delete mode 100644 demos/bbm_auxiliary/bbm_final.png delete mode 100644 demos/bbm_auxiliary/bbm_init.png delete mode 100644 demos/bbm_auxiliary/invariant_errors.png delete mode 100644 demos/bbm_auxiliary/invariants.png create mode 100644 docs/source/demos/bbm_aux_errors.png create mode 100644 docs/source/demos/bbm_aux_final.png create mode 100644 docs/source/demos/bbm_aux_init.png create mode 100644 docs/source/demos/bbm_aux_invariants.png delete mode 100644 docs/source/demos/bbm_final.png create mode 100644 docs/source/demos/bbm_hamiltonian_errors.png create mode 100644 docs/source/demos/bbm_hamiltonian_final.png create mode 100644 docs/source/demos/bbm_hamiltonian_init.png 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z_Vm*_J`WeyN4H-PMgS0^PiH{T7qyV!r6zr}D%jTD9SO|2Jm_7U9neJuoxcSajZJ=_ z;&g4=0LC%{oFpjU7z~lwpTc?Z_d9{vxFl3a3evbB1(Xb`X#YR@4*_Jq%$?yl;1Y$V T1V`k7u^}aSHM#t|rmy}NX;oz_ diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index 7e5b6e50..288a9f72 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -1,5 +1,5 @@ -Hamiltonian-structure-preserving implementation of the Benjamin-Bona-Mahoney equation -===================================================================================== +Mixed Hamiltonian-preserving formulation of the Benjamin-Bona-Mahoney equation +============================================================================== This demo solves the Benjamin-Bona-Mahony equation: @@ -20,18 +20,18 @@ BBM is known to have a Hamiltonian structure, and there are several canonical po I_3 & = \int \frac{u^2}{2} + \frac{u^3}{6} \, \mathrm{d}x The BBM invariants are the total momentum :math:`I_1`, the :math:`H^1`-energy -norm :math:`I_2`, and the Hamiltonian :math:`I_3`. -The Hamiltonian variational formulation reads +:math:`I_2`, and the Hamiltonian :math:`I_3`. +The Hamiltonian mixed variational formulation reads .. math:: - (\partial_t u + \partial_x \tilde{w}_H, v)_{H^1} & = 0 + (\partial_t u, v)_{H^1} - (\tilde{w}_H, \partial_x v_H)_{L^2} & = 0 - (\tilde{w}_H, v_H)_{H^1} & = \langle \frac{\delta I_3}{\delta u}, v_H \rangle + (\tilde{w}_H, v_H)_{L^2} & = \langle \frac{\delta I_3}{\delta u}, v_H \rangle For all test functions :math:`v, v_H` in a suitable function space. The numerical scheme in this demo introduces -the :math:`H^1`-Riesz representative of the Fréchet derivative of the +the :math:`L^2`-Riesz representative of the Fréchet derivative of the Hamiltonian :math:`\frac{\delta I_3}{\delta u}` as the auxiliary variable :math:`\tilde{w}_H`. @@ -52,8 +52,8 @@ Firedrake, Irksome, and other imports:: from firedrake import (Constant, Function, FunctionSpace, PeriodicIntervalMesh, SpatialCoordinate, TestFunction, TrialFunction, - assemble, derivative, dx, errornorm, exp, grad, inner, - interpolate, norm, plot, project, replace, solve, split + assemble, derivative, dx, exp, grad, inner, + norm, plot, project, replace, solve, split ) from irksome import Dt, GalerkinTimeStepper, TimeQuadratureLabel @@ -88,11 +88,11 @@ Next, we define the domain and the exact solution :: This sets up the function space for the unknown :math:`u` and auxiliary variable :math:`\tilde{w}_H`:: - space_deg = 3 + space_deg = 1 time_deg = 1 - V = FunctionSpace(msh, "Hermite", space_deg) - Z = V * V + V = FunctionSpace(msh, "CG", space_deg) + Z = V*V We next define the BBM invariants. Again, the discrete formulation preserves :math:`I_1` and :math:`I_3` up to solver tolerances and roundoff errors, @@ -119,27 +119,28 @@ initial condition for the auxiliary variable. We need to find :math:`\tilde{w}_ :: - uwH = Function(Z) - u0, wH0 = uwH.subfunctions + uw = Function(Z) + u0, w0 = uw.subfunctions v = TestFunction(V) w = TrialFunction(V) - a = h1inner(w, v) * dx + a0 = inner(w, v) * dx + a1 = h1inner(w, v) * dx dHdu = derivative(I3(u0), u0, v) - solve(a == h1inner(uexact, v)*dx, u0) - solve(a == dHdu, wH0) + solve(a1 == h1inner(uexact, v)*dx, u0) + solve(a0 == dHdu, w0) Visualize the initial condition:: fig, axes = plt.subplots(1) - plot(Function(FunctionSpace(msh, "CG", 1)).interpolate(u0), axes=axes) + plot(u0, axes=axes) axes.set_title("Initial condition") axes.set_xlabel("x") axes.set_ylabel("u") - plt.savefig("bbm_init.png") + plt.savefig("bbm_aux_init.png") -.. figure:: bbm_init.png +.. figure:: bbm_aux_init.png :align: center Create time quadrature labels:: @@ -153,10 +154,9 @@ Create time quadrature labels:: This tags several of the terms with a low-order time integration scheme, but forces a higher-order method on the nonlinear term:: - u, wH = split(uwH) + u, w = split(uw) v, vH = split(TestFunction(Z)) - - Flow = h1inner(Dt(u) + wH.dx(0), v) * dx + h1inner(wH, vH) * dx + Flow = h1inner(Dt(u), v) * dx + inner(w.dx(0), v)*dx + inner(w, vH)*dx Fhigh = replace(dHdu, {u0: u}) F = Llow(Flow) - Lhigh(Fhigh(vH)) @@ -165,8 +165,7 @@ This sets up the cPG time stepper. There are two fields in the unknown, we indicate the second one is an auxiliary and hence to be discretized in the DG test space instead by passing the `aux_indices` keyword:: - stepper = GalerkinTimeStepper( - F, time_deg, t, dt, uwH, aux_indices=[1]) + stepper = GalerkinTimeStepper(F, time_deg, t, dt, uw, aux_indices=[1]) UFL expressions for the invariants, which we are going to track as we go through time steps:: @@ -201,7 +200,7 @@ Visualize invariant preservation:: axes.set_xlabel("Time") axes.set_ylabel("I(t)") axes.legend() - plt.savefig("invariants.png") + plt.savefig("bbm_aux_invariants.png") axes.clear() for i in (0, 1, 2): @@ -210,22 +209,22 @@ Visualize invariant preservation:: axes.set_xlabel("Time") axes.set_ylabel("|1-I/I(0)|") axes.legend() - plt.savefig("invariant_errors.png") + plt.savefig("bbm_aux_errors.png") -.. figure:: invariants.png +.. figure:: bbm_aux_invariants.png :align: center -.. figure:: invariant_errors.png +.. figure:: bbm_aux_errors.png :align: center 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demos/demo_bbm.py -and with a Galerkin-in-Time approach, in standard form or with auxiliary variables for alternate conservation properties: +and with a Galerkin-in-Time approach, in standard form, in Hamiltonian form, or with auxiliary variables for alternate conservation properties: .. toctree:: :maxdepth: 1 demos/demo_bbm_galerkin.py + demos/demo_bbm_hamiltonian.py demos/demo_bbm_aux.py Finally, if you feel you must bypass the :py:class:`.TimeStepper` From 056dbe772eed2ca0c3f01def61373dfd9b4bc744 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 24 Oct 2025 11:59:37 +0100 Subject: [PATCH 2/7] cleanup --- demos/bbm_auxiliary/demo_bbm_aux.py.rst | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index 288a9f72..68fb45b9 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -52,8 +52,7 @@ Firedrake, Irksome, and other imports:: from firedrake import (Constant, Function, FunctionSpace, PeriodicIntervalMesh, SpatialCoordinate, TestFunction, TrialFunction, - assemble, derivative, dx, exp, grad, inner, - norm, plot, project, replace, solve, split + assemble, derivative, dx, exp, grad, inner, plot, replace, solve, split ) from irksome import Dt, GalerkinTimeStepper, TimeQuadratureLabel From a22bdfa2c96ef7d2ae920c2a32d7c2888f7dd861 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 24 Oct 2025 12:23:55 +0100 Subject: [PATCH 3/7] rescale I2 --- demos/bbm_auxiliary/demo_bbm_aux.py.rst | 4 ++-- docs/source/demos/bbm_aux_invariants.png | Bin 12360 -> 12624 bytes .../demos/bbm_hamiltonian_invariants.png | Bin 12360 -> 12624 bytes 3 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index 68fb45b9..eda57d31 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -15,7 +15,7 @@ BBM is known to have a Hamiltonian structure, and there are several canonical po I_1 & = \int u \, \mathrm{d}x - I_2 & = \int u^2 + (u_x)^2 \, \mathrm{d}x + I_2 & = \int \frac{u^2 + (u_x)^2}{2} \, \mathrm{d}x I_3 & = \int \frac{u^2}{2} + \frac{u^3}{6} \, \mathrm{d}x @@ -104,7 +104,7 @@ but :math:`I_2` is preserved up to a bounded oscillation :: return u * dx def I2(u): - return h1inner(u, u) * dx + return 0.5 * h1inner(u, u) * dx def I3(u): return (u**2 / 2 + u**3 / 6) * dx diff --git a/docs/source/demos/bbm_aux_invariants.png b/docs/source/demos/bbm_aux_invariants.png index 92dac3aed1b1ca6c7e9e2810ab4e3a9463ecf267..ce0da0d583e3099c6ad4d9480eb2a6b8b3c46a46 100644 GIT binary patch literal 12624 zcmeHt2UJw)*5w6M6jWLf5hR#sl~5>%WKb~yl9M1oM6v|Q8Ct~zid`fHib6>RMUYfT z5gQZ)BqyQBNY1f{#hmNw_4k{3|9kV+nl-a#X8kS$R^6)m-S0bRpS|}vFVt0U>|^3& zLJ(x1;?1j?2tpr#Aawn^cfl)SP2E4>A1Rk>w_UUxEL_}8oy-vxQy0ev4lWO@?;Ue9 zcXGCNuouOgKZg+$JZ9zM;^-_TENu55S702REQR$qtfb*v_Bh_ucSaBvQ}hoVK|a|U 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z;tROyvamHP_2a{cN?ipIN$n08r;$kDJ8)`Gy$x2QG4r^`kB_N0HZ?uvLyafkfH4j0 zI{-vW9|X1B@8K*XgBwo+la}D7$kAT~!VIsHK5`5nDjVG0y1Ns`Ta6T9F|m@J!dpTY z(5+x|1Ai15M)w9_)Me*t))We@t%Z4*8w-#XuOIElQEY;Y?*iW#YI1?XtpNoiO~u1E7MHTL+c3|JZQ=9TCZS`#solO Date: Fri, 24 Oct 2025 12:28:19 +0100 Subject: [PATCH 4/7] cleanup --- demos/bbm_auxiliary/demo_bbm_aux.py.rst | 17 ++++++++--------- 1 file changed, 8 insertions(+), 9 deletions(-) diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index eda57d31..e8f3a505 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -84,14 +84,14 @@ Next, we define the domain and the exact solution :: uexact = 3 * c**2 / (1-c**2) \ * sech(0.5 * (c * x - c * t / (1 - c ** 2) + delta))**2 -This sets up the function space for the unknown :math:`u` and +This sets up the mixed function space for the unknown :math:`u` and auxiliary variable :math:`\tilde{w}_H`:: space_deg = 1 time_deg = 1 V = FunctionSpace(msh, "CG", space_deg) - Z = V*V + Z = V * V We next define the BBM invariants. Again, the discrete formulation preserves :math:`I_1` and :math:`I_3` up to solver tolerances and roundoff errors, @@ -109,12 +109,12 @@ but :math:`I_2` is preserved up to a bounded oscillation :: def I3(u): return (u**2 / 2 + u**3 / 6) * dx -We project the initial condition on :math:`u`, but we also need a consistent +We project the initial condition on :math:`u` in the :math:`H^1` norm, but we also need a consistent initial condition for the auxiliary variable. We need to find :math:`\tilde{w}_H \in V` such that .. math:: - (\tilde{w}_H, v)_{H^1} = \langle \frac{\delta I_3}{\delta u}, v \rangle \text{ for all } v \in V + (\tilde{w}_H, v)_{L^2} = \langle \frac{\delta I_3}{\delta u}, v \rangle \text{ for all } v \in V :: @@ -123,12 +123,11 @@ initial condition for the auxiliary variable. We need to find :math:`\tilde{w}_ v = TestFunction(V) w = TrialFunction(V) - a0 = inner(w, v) * dx - a1 = h1inner(w, v) * dx - dHdu = derivative(I3(u0), u0, v) - solve(a1 == h1inner(uexact, v)*dx, u0) - solve(a0 == dHdu, w0) + solve(h1inner(w, v)*dx == h1inner(uexact, v)*dx, u0) + + dHdu = derivative(I3(u0), u0, v) + solve(inner(w, v)*dx == dHdu, w0) Visualize the initial condition:: From 9b1d1013d7a597ab5454e594d08fad55bf6ab446 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 24 Oct 2025 14:00:13 +0100 Subject: [PATCH 5/7] cleanup --- demos/bbm_auxiliary/demo_bbm_aux.py.rst | 41 +++++++++++++------------ 1 file changed, 22 insertions(+), 19 deletions(-) diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index e8f3a505..ecb0b77a 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -21,31 +21,34 @@ BBM is known to have a Hamiltonian structure, and there are several canonical po The BBM invariants are the total momentum :math:`I_1`, the :math:`H^1`-energy :math:`I_2`, and the Hamiltonian :math:`I_3`. -The Hamiltonian mixed variational formulation reads + +Standard Gauss-Legendre and continuous Petrov-Galerkin (cPG) methods conserve +the first two invariants exactly (up to roundoff and solver tolerances). They +do quite well, but are inexact for the cubic one. In this demo, we consider a +mixed Hamiltonian formulation that preserves the third invariant at the expense +of the second. The mixed formulation solves .. math:: - (\partial_t u, v)_{H^1} - (\tilde{w}_H, \partial_x v_H)_{L^2} & = 0 + (\partial_t u, v)_{H^1} & = (\tilde{w}_H, \partial_x v)_{L^2} (\tilde{w}_H, v_H)_{L^2} & = \langle \frac{\delta I_3}{\delta u}, v_H \rangle -For all test functions :math:`v, v_H` in a suitable function space. -The numerical scheme in this demo introduces -the :math:`L^2`-Riesz representative of the Fréchet derivative of the -Hamiltonian :math:`\frac{\delta I_3}{\delta u}` -as the auxiliary variable :math:`\tilde{w}_H`. +for all test functions :math:`v, v_H` in a suitable function space :math:`V \times V`. +In this demo we choose to discretize :math:`V` with :math:`C^0` +Lagrange elements by introducing the auxiliary variable :math:`\tilde{w}_H \in V` +that holds the :math:`L^2`-Riesz representative of the Fréchet derivative of the +Hamiltonian :math:`\frac{\delta I_3}{\delta u}`. -Standard Gauss-Legendre and continuous Petrov-Galerkin (cPG) methods conserve -the first two invariants exactly (up to roundoff and solver tolerances). They -do quite well, but are inexact for the cubic one. Here, we consider the -reformulation in Andrews and Farrell, "Enforcing conservation laws and dissipation -inequalities numerically via auxiliary variables" (`arXiv:2407.11904 `_, to appear -in SIAM J. Scientific Computing) that preserves the third invariant at -the expense of the second. This method has an auxiliary variable in the system -and requires a continuously differentiable spatial discretization (1D Hermite -elements in this case). The time discretization puts the main unknown in a -continuous space and the auxiliary variable in a discontinuous one. See -equation (7.17) of Boris Andrews' thesis for the particular formulation. +.. note:: + + Here, we consider the framework in Andrews and Farrell, "Enforcing conservation laws and dissipation + inequalities numerically via auxiliary variables" (`arXiv:2407.11904 `_, to appear + in SIAM J. Scientific Computing). Their method has an auxiliary variable in the system + and requires a continuously differentiable spatial discretization (1D Hermite + elements in their case). The time discretization puts the main unknown in a + continuous space and the auxiliary variable in a discontinuous one. See + equation (7.17) of Boris Andrews' thesis for their particular formulation. Firedrake, Irksome, and other imports:: @@ -154,7 +157,7 @@ but forces a higher-order method on the nonlinear term:: u, w = split(uw) v, vH = split(TestFunction(Z)) - Flow = h1inner(Dt(u), v) * dx + inner(w.dx(0), v)*dx + inner(w, vH)*dx + Flow = h1inner(Dt(u), v) * dx - inner(w, v.dx(0))*dx + inner(w, vH)*dx Fhigh = replace(dHdu, {u0: u}) F = Llow(Flow) - Lhigh(Fhigh(vH)) From 3e933e83d5a4ae86255f5142f937c271ef80e7f4 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 24 Oct 2025 14:41:46 +0100 Subject: [PATCH 6/7] add the primal demo --- .../bbm_auxiliary/demo_bbm_hamiltonian.py.rst | 203 ++++++++++++++++++ 1 file changed, 203 insertions(+) create mode 100644 demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst diff --git a/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst b/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst new file mode 100644 index 00000000..319518fa --- /dev/null +++ b/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst @@ -0,0 +1,203 @@ +Primal Hamiltonian-preserving formulation of the Benjamin-Bona-Mahoney equation +=============================================================================== + +This demo solves the Benjamin-Bona-Mahony equation: + +.. math:: + + u_t - u_{txx} + u_x + u u_x = 0 + +posed on a bounded interval with periodic boundaries. + +BBM is known to have a Hamiltonian structure, and there are several canonical polynomial invariants: + +.. math:: + + I_1 & = \int u \, \mathrm{d}x + + I_2 & = \int \frac{u^2 + (u_x)^2}{2} \, \mathrm{d}x + + I_3 & = \int \frac{u^2}{2} + \frac{u^3}{6} \, \mathrm{d}x + +The BBM invariants are the total momentum :math:`I_1`, the :math:`H^1`-energy +:math:`I_2`, and the Hamiltonian :math:`I_3`. + +Standard Gauss-Legendre and continuous Petrov-Galerkin (cPG) methods conserve +the first two invariants exactly (up to roundoff and solver tolerances). They +do quite well, but are inexact for the cubic one. +In this demo, we consider a primal Hamiltonian formulation that preserves the third invariant at +the expense of the second. The problem is to find :math:`u \in V` such that + +.. math:: + + \langle \partial_t \frac{\delta I_2}{\delta u}, v \rangle = \langle \frac{\delta I_3}{\delta u}, \partial_x v \rangle + +for all test functions :math:`v \in V`. +Here :math:`\frac{\delta I_3}{\delta u} : V \to \mathbb{R}` is the Fréchet derivative of the +Hamiltonian. A primal conforming discretization requires that :math:`\partial_x V \subset V`, +hence we use :math:`C^1` Hermite elements. A mixed reformulation using +:math:`C^0` Lagrange elements with the same conservation properties is found in :doc:`this demo `. + +Firedrake, Irksome, and other imports:: + + from firedrake import (Constant, Function, FunctionSpace, + PeriodicIntervalMesh, SpatialCoordinate, TestFunction, TrialFunction, + assemble, derivative, dx, exp, grad, inner, plot, solve, + ) + + from irksome import Dt, GalerkinTimeStepper, TimeQuadratureLabel + + import matplotlib.pyplot as plt + import numpy + +Next, we define the domain and the exact solution :: + + N = 8000 + L = 100 + h = L / N + msh = PeriodicIntervalMesh(N, L) + x, = SpatialCoordinate(msh) + + t = Constant(0) + inv_dt = N // (10 * L) + tfinal = 18 + Nt = tfinal * inv_dt + dt = Constant(tfinal / Nt) + + c = Constant(0.5) + center = Constant(40.0) + delta = -c * center + + def sech(x): + return 2 / (exp(x) + exp(-x)) + + uexact = 3 * c**2 / (1-c**2) \ + * sech(0.5 * (c * x - c * t / (1 - c ** 2) + delta))**2 + +This sets up the function space for the unknown :math:`u` :: + + space_deg = 3 + time_deg = 1 + + V = FunctionSpace(msh, "Hermite", space_deg) + +We next define the BBM invariants. Again, the discrete formulation preserves +:math:`I_1` and :math:`I_3` up to solver tolerances and roundoff errors, +but :math:`I_2` is preserved up to a bounded oscillation :: + + def h1inner(u, v): + return inner(u, v) + inner(grad(u), grad(v)) + + def I1(u): + return u * dx + + def I2(u): + return 0.5 * h1inner(u, u) * dx + + def I3(u): + return (u**2 / 2 + u**3 / 6) * dx + +We project the initial condition on :math:`u`. :: + + u = Function(V) + + v = TestFunction(V) + w = TrialFunction(V) + + a = h1inner(w, v) * dx + + solve(a == h1inner(uexact, v)*dx, u) + +Visualize the initial condition:: + + fig, axes = plt.subplots(1) + plot(Function(FunctionSpace(msh, "CG", space_deg)).interpolate(u), axes=axes) + axes.set_title("Initial condition") + axes.set_xlabel("x") + axes.set_ylabel("u") + plt.savefig("bbm_hamiltonian_init.png") + +.. figure:: bbm_hamiltonian_init.png + :align: center + +Create time quadrature labels:: + + time_order_low = 2 * (time_deg - 1) + time_order_high = 3 * time_deg - 1 + + Llow = TimeQuadratureLabel(time_order_low) + Lhigh = TimeQuadratureLabel(time_order_high) + +This tags several of the terms with a low-order time integration scheme, +but forces a higher-order method on the nonlinear term:: + + dHdu = derivative(I3(u), u, v) + Flow = h1inner(Dt(u), v)*dx + + F = Llow(Flow) - Lhigh(dHdu(v.dx(0))) + +This sets up the cPG time stepper. :: + + stepper = GalerkinTimeStepper(F, time_deg, t, dt, u) + +UFL expressions for the invariants, which we are going to track as we go +through time steps:: + + times = [float(t)] + functionals = (I1(u), I2(u), I3(u)) + invariants = [tuple(map(assemble, functionals))] + +Do the time-stepping:: + + for _ in range(Nt): + stepper.advance() + + invariants.append(tuple(map(assemble, functionals))) + + i1, i2, i3 = invariants[-1] + t.assign(float(t) + float(dt)) + times.append(float(t)) + + print(f'{float(t):.15f}, {i1:.15f}, {i2:.15f}, {i3:.15f}') + +Visualize invariant preservation:: + + axes.clear() + invariants = numpy.array(invariants) + + lbls = ("I1", "I2", "I3") + + for i in (0, 1, 2): + plt.plot(times, invariants[:, i], label=lbls[i]) + axes.set_title("Invariants over time") + axes.set_xlabel("Time") + axes.set_ylabel("I(t)") + axes.legend() + plt.savefig("bbm_hamiltonian_invariants.png") + axes.clear() + + for i in (0, 1, 2): + plt.plot(times, 1.0 - invariants[:, i]/invariants[0, i], label=lbls[i]) + axes.set_title("Relative error in invariants over time") + axes.set_xlabel("Time") + axes.set_ylabel("|1-I/I(0)|") + axes.legend() + plt.savefig("bbm_hamiltonian_errors.png") + +.. figure:: bbm_hamiltonian_invariants.png + :align: center + +.. figure:: bbm_hamiltonian_errors.png + :align: center + +Visualize the solution at final time step:: + + axes.clear() + plot(Function(FunctionSpace(msh, "CG", space_deg)).interpolate(u), axes=axes) + axes.set_title(f"Solution at time {tfinal}") + axes.set_xlabel("x") + axes.set_ylabel("u") + plt.savefig("bbm_hamiltonian_final.png") + +.. figure:: bbm_hamiltonian_final.png + :align: center From 40e1cbc19140cf2b511c56d425b6120cde7005fd Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Mon, 27 Oct 2025 13:17:54 +0000 Subject: [PATCH 7/7] Tighten solver tolerance --- demos/bbm_auxiliary/demo_bbm_aux.py.rst | 5 ++++- demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst | 7 +++++-- 2 files changed, 9 insertions(+), 3 deletions(-) diff --git a/demos/bbm_auxiliary/demo_bbm_aux.py.rst b/demos/bbm_auxiliary/demo_bbm_aux.py.rst index ecb0b77a..61a7bbed 100644 --- a/demos/bbm_auxiliary/demo_bbm_aux.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_aux.py.rst @@ -166,7 +166,10 @@ This sets up the cPG time stepper. There are two fields in the unknown, we indicate the second one is an auxiliary and hence to be discretized in the DG test space instead by passing the `aux_indices` keyword:: - stepper = GalerkinTimeStepper(F, time_deg, t, dt, uw, aux_indices=[1]) + sparams = {"snes_atol": 0, "snes_rtol": 1E-14} + stepper = GalerkinTimeStepper(F, time_deg, t, dt, uw, + aux_indices=[1], + solver_parameters=sparams) UFL expressions for the invariants, which we are going to track as we go through time steps:: diff --git a/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst b/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst index 319518fa..60f82c3d 100644 --- a/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst +++ b/demos/bbm_auxiliary/demo_bbm_hamiltonian.py.rst @@ -137,8 +137,11 @@ but forces a higher-order method on the nonlinear term:: F = Llow(Flow) - Lhigh(dHdu(v.dx(0))) This sets up the cPG time stepper. :: - - stepper = GalerkinTimeStepper(F, time_deg, t, dt, u) + + sparams = {"snes_atol": 0, "snes_rtol": 1E-14} + stepper = GalerkinTimeStepper(F, time_deg, t, dt, u, + solver_parameters=sparams) + UFL expressions for the invariants, which we are going to track as we go through time steps::