|
| 1 | +from pySDC.core.Errors import ParameterError |
| 2 | +from pySDC.projects.DAE.sweepers.fully_implicit_DAE import fully_implicit_DAE |
| 3 | + |
| 4 | + |
| 5 | +class SemiImplicitDAE(fully_implicit_DAE): |
| 6 | + r""" |
| 7 | + Custom sweeper class to implement SDC for solving semi-explicit DAEs of the form |
| 8 | +
|
| 9 | + .. math:: |
| 10 | + u' = f(u, z, t), |
| 11 | +
|
| 12 | + .. math:: |
| 13 | + 0 = g(u, z, t) |
| 14 | +
|
| 15 | + with :math:`u(t), u'(t) \in\mathbb{R}^{N_d}` the differential variables and their derivates, |
| 16 | + algebraic variables :math:`z(t) \in\mathbb{R}^{N_a}`, :math:`f(u, z, t) \in \mathbb{R}^{N_d}`, |
| 17 | + and :math:`g(u, z, t) \in \mathbb{R}^{N_a}`. :math:`N = N_d + N_a` is the dimension of the whole |
| 18 | + system of DAEs. |
| 19 | +
|
| 20 | + It solves a collocation problem of the form |
| 21 | +
|
| 22 | + .. math:: |
| 23 | + U = f(\vec{U}_0 + \Delta t (\mathbf{Q} \otimes \mathbf{I}_{n_d}) \vec{U}, \vec{z}, \tau), |
| 24 | +
|
| 25 | + .. math:: |
| 26 | + 0 = g(\vec{U}_0 + \Delta t (\mathbf{Q} \otimes \mathbf{I}_{n_d}) \vec{U}, \vec{z}, \tau), |
| 27 | +
|
| 28 | + where |
| 29 | + |
| 30 | + - :math:`\tau=(\tau_1,..,\tau_M) in \mathbb{R}^M` the vector of collocation nodes, |
| 31 | + - :math:`\vec{U}_0 = (u_0,..,u_0) \in \mathbb{R}^{MN_d}` the vector of initial condition spread to each node, |
| 32 | + - spectral integration matrix :math:`\mathbf{Q} \in \mathbb{R}^{M \times M}`, |
| 33 | + - :math:`\vec{U}=(U_1,..,U_M) \in \mathbb{R}^{MN_d}` the vector of unknown derivatives of differential variables |
| 34 | + :math:`U_m \approx U(\tau_m) = u'(\tau_m) \in \mathbb{R}^{N_d}`, |
| 35 | + - :math:`\vec{z}=(z_1,..,z_M) \in \mathbb{R}^{MN_a}` the vector of unknown algebraic variables |
| 36 | + :math:`z_m \approx z(\tau_m) \in \mathbb{R}^{N_a}`, |
| 37 | + - and identity matrix :math:`\mathbf{I}_{N_d} \in \mathbb{R}^{N_d \times N_d}`. |
| 38 | +
|
| 39 | + This sweeper treats the differential and the algebraic variables differently by only integrating the differential |
| 40 | + components. Solving the nonlinear system, :math:`{U,z}` are the unknowns. |
| 41 | +
|
| 42 | + The sweeper implementation is based on the ideas mentioned in the KDC publication [1]_. |
| 43 | +
|
| 44 | + Parameters |
| 45 | + ---------- |
| 46 | + params : dict |
| 47 | + Parameters passed to the sweeper. |
| 48 | +
|
| 49 | + Attributes |
| 50 | + ---------- |
| 51 | + QI : np.2darray |
| 52 | + Implicit Euler integration matrix. |
| 53 | +
|
| 54 | + References |
| 55 | + ---------- |
| 56 | + .. [1] J. Huang, J. Jun, M. L. Minion. Arbitrary order Krylov deferred correction methods for differential algebraic |
| 57 | + equation. J. Comput. Phys. Vol. 221 No. 2 (2007). |
| 58 | +
|
| 59 | + Note |
| 60 | + ---- |
| 61 | + The right-hand side of the problem DAE classes using this sweeper has to be exactly implemented in the way, the |
| 62 | + semi-explicit DAE is defined. Define :math:`\vec{x}=(y, z)^T`, :math:`F(\vec{x})=(f(\vec{x}), g(\vec{x}))`, and the |
| 63 | + matrix |
| 64 | +
|
| 65 | + .. math:: |
| 66 | + A = \begin{matrix} |
| 67 | + I & 0 \\ |
| 68 | + 0 & 0 |
| 69 | + \end{matrix} |
| 70 | +
|
| 71 | + then, the problem can be reformulated as |
| 72 | +
|
| 73 | + .. math:: |
| 74 | + A\vec{x}' = F(\vec{x}). |
| 75 | +
|
| 76 | + Then, setting :math:`F_{new}(\vec{x}, \vec{x}') = A\vec{x}' - F(\vec{x})` defines a DAE of fully-implicit form |
| 77 | +
|
| 78 | + .. math:: |
| 79 | + 0 = F_{new}(\vec{x}, \vec{x}'). |
| 80 | +
|
| 81 | + Hence, the method ``eval_f`` of problem DAE classes of semi-explicit form implements the right-hand side in the way of |
| 82 | + returning :math:`F(\vec{x})`, whereas ``eval_f`` of problem classes of fully-implicit form return the right-hand side |
| 83 | + :math:`F_{new}(\vec{x}, \vec{x}')`. |
| 84 | + """ |
| 85 | + |
| 86 | + def __init__(self, params): |
| 87 | + """Initialization routine""" |
| 88 | + |
| 89 | + if 'QI' not in params: |
| 90 | + params['QI'] = 'IE' |
| 91 | + |
| 92 | + # call parent's initialization routine |
| 93 | + super().__init__(params) |
| 94 | + |
| 95 | + msg = f"Quadrature type {self.params.quad_type} is not implemented yet. Use 'RADAU-RIGHT' instead!" |
| 96 | + if self.coll.left_is_node: |
| 97 | + raise ParameterError(msg) |
| 98 | + |
| 99 | + self.QI = self.get_Qdelta_implicit(coll=self.coll, qd_type=self.params.QI) |
| 100 | + |
| 101 | + def integrate(self): |
| 102 | + r""" |
| 103 | + Returns the solution by integrating its gradient (fundamental theorem of calculus) at each collocation node. |
| 104 | + ``level.f`` stores the gradient of solution ``level.u``. |
| 105 | +
|
| 106 | + Returns |
| 107 | + ------- |
| 108 | + me : list of lists |
| 109 | + Integral of the gradient at each collocation node. |
| 110 | + """ |
| 111 | + |
| 112 | + # get current level and problem description |
| 113 | + L = self.level |
| 114 | + P = L.prob |
| 115 | + M = self.coll.num_nodes |
| 116 | + |
| 117 | + me = [] |
| 118 | + for m in range(1, M + 1): |
| 119 | + # new instance of dtype_u, initialize values with 0 |
| 120 | + me.append(P.dtype_u(P.init, val=0.0)) |
| 121 | + for j in range(1, M + 1): |
| 122 | + me[-1].diff[:] += L.dt * self.coll.Qmat[m, j] * L.f[j].diff[:] |
| 123 | + |
| 124 | + return me |
| 125 | + |
| 126 | + def update_nodes(self): |
| 127 | + r""" |
| 128 | + Updates the values of solution ``u`` and their gradient stored in ``f``. |
| 129 | + """ |
| 130 | + |
| 131 | + L = self.level |
| 132 | + P = L.prob |
| 133 | + |
| 134 | + # only if the level has been touched before |
| 135 | + assert L.status.unlocked |
| 136 | + M = self.coll.num_nodes |
| 137 | + |
| 138 | + integral = self.integrate() |
| 139 | + # build the rest of the known solution u_0 + del_t(Q - Q_del)U_k |
| 140 | + for m in range(1, M + 1): |
| 141 | + for j in range(1, m + 1): |
| 142 | + integral[m - 1].diff[:] -= L.dt * self.QI[m, j] * L.f[j].diff[:] |
| 143 | + integral[m - 1].diff[:] += L.u[0].diff |
| 144 | + |
| 145 | + # do the sweep |
| 146 | + for m in range(1, M + 1): |
| 147 | + u_approx = P.dtype_u(integral[m - 1]) |
| 148 | + for j in range(1, m): |
| 149 | + u_approx.diff[:] += L.dt * self.QI[m, j] * L.f[j].diff[:] |
| 150 | + |
| 151 | + def implSystem(unknowns): |
| 152 | + """ |
| 153 | + Build implicit system to solve in order to find the unknowns. |
| 154 | +
|
| 155 | + Parameters |
| 156 | + ---------- |
| 157 | + unknowns : dtype_u |
| 158 | + Unknowns of the system. |
| 159 | +
|
| 160 | + Returns |
| 161 | + ------- |
| 162 | + sys : |
| 163 | + System to be solved as implicit function. |
| 164 | + """ |
| 165 | + |
| 166 | + unknowns_mesh = P.dtype_f(unknowns) |
| 167 | + |
| 168 | + local_u_approx = P.dtype_u(u_approx) |
| 169 | + local_u_approx.diff[:] += L.dt * self.QI[m, m] * unknowns_mesh.diff[:] |
| 170 | + local_u_approx.alg[:] = unknowns_mesh.alg[:] |
| 171 | + |
| 172 | + sys = P.eval_f(local_u_approx, unknowns_mesh, L.time + L.dt * self.coll.nodes[m - 1]) |
| 173 | + return sys |
| 174 | + |
| 175 | + u0 = P.dtype_u(P.init) |
| 176 | + u0.diff[:], u0.alg[:] = L.f[m].diff[:], L.u[m].alg[:] |
| 177 | + u_new = P.solve_system(implSystem, u0, L.time + L.dt * self.coll.nodes[m - 1]) |
| 178 | + # ---- update U' and z ---- |
| 179 | + L.f[m].diff[:] = u_new.diff[:] |
| 180 | + L.u[m].alg[:] = u_new.alg[:] |
| 181 | + |
| 182 | + # Update solution approximation |
| 183 | + integral = self.integrate() |
| 184 | + for m in range(M): |
| 185 | + L.u[m + 1].diff[:] = L.u[0].diff[:] + integral[m].diff[:] |
| 186 | + |
| 187 | + # indicate presence of new values at this level |
| 188 | + L.status.updated = True |
| 189 | + |
| 190 | + return None |
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