tindex.rst - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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tindex.rst (9155B)
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1 .. include:: ../../global.txt
2
3 .. _sec-verif:
4
5 Verification
6 ============
7
8 Two types of errors may be distinguished: modeling errors and numerical errors. Modeling
9 errors arise from not solving the right equations. Numerical errors result from not
10 solving the equations right. The assessment of modeling errors is *validation*, whereas
11 the assessment of numerical errors is called *verification*... Validation makes sense
12 only after verification, otherwise agreement between measured and computed results may
13 well be fortuitous.
14
15 P. Wesseling, (2001) *Principles of Computational Fluid Dynamics*, pp. 560--561
16 :cite:`Wesseling`
17
18 Verification is the essentially mathematical task of checking that the predictions of the
19 numerical code are close to the predictions of a continuum model, the one which the
20 numerical code claims to approximate. It is a crucial task for a code as complicated as
21 PISM. In verification there is no comparison between model output and observations of
22 nature. Instead, one compares exact solutions of the continuum model, in circumstances in
23 which they are available, to their numerical approximations.
24
25 Reference :cite:`Roache` gives a broad discussion of verification and validation in
26 computational fluid dynamics. See :cite:`BLKCB` and :cite:`BBL` for discussion of verification
27 issues for the isothermal and thermomechanically coupled shallow ice approximation (SIA),
28 respectively, and for exact solutions to these models, and :cite:`BBssasliding`,
29 :cite:`SchoofStream` for verification using an exact solution to the SSA equations for ice
30 streams.
31
32 In PISM there is a separate executable ``pismv`` which is used for SIA-related
33 verification, and there are additional scripts for SSA-related verification. The source
34 codes which are verified by ``pismv`` are, however, exactly the same source files as are
35 run by the normal PISM executable ``pismr``. In technical terms, ``pismv`` runs a derived
36 class of the PISM base class.
37
38 .. list-table:: Exact solutions for verification
39 :header-rows: 1
40 :name: tab-tests
41 :widths: 1,4,2,4
42
43 * - Test
44 - Continuum model tested
45 - Reference
46 - Comments
47 * - A
48 - isothermal SIA, steady, flat bed, constant accumulation
49 - :cite:`BLKCB`
50 -
51 * - B
52 - isothermal SIA, flat bed, zero accumulation
53 - :cite:`BLKCB`, :cite:`Halfar83`
54 - similarity solution
55 * - C
56 - isothermal SIA, flat bed, growing accumulation
57 - :cite:`BLKCB`
58 - similarity solution
59 * - D
60 - isothermal SIA, flat bed, oscillating accumulation
61 - :cite:`BLKCB`
62 - uses compensatory accumulation
63 * - E
64 - isothermal SIA; as A, but with sliding in a sector
65 - :cite:`BLKCB`
66 - uses compensatory accumulation
67 * - F
68 - thermomechanically coupled SIA (mass and energy conservation), steady, flat bed
69 - :cite:`BB`, :cite:`BBL`
70 - uses compensatory accumulation and heating
71 * - G
72 - thermomechanically coupled SIA; as F but with oscillating accumulation
73 - :cite:`BB`, :cite:`BBL`
74 - ditto
75 * - H
76 - bed deformation coupled with isothermal SIA
77 - :cite:`BLKfastearth`
78 - joined similarity solution
79 * - I
80 - stream velocity computation using SSA (plastic till)
81 - :cite:`SchoofStream`, :cite:`BBssasliding`
82 -
83 * - J
84 - shelf velocity computation using SSA
85 - (source code)
86 -
87 * - K
88 - pure conduction in ice and bedrock
89 - :cite:`BuelerTestK`
90 -
91 * - L
92 - isothermal SIA, steady, non-flat bed
93 - (source code)
94 - numerical ODE solution
95
96 .. csv-table:: Canonical PISM verification runs using the exact solutions listed in
97 :numref:`tab-tests`.
98 :header: Test, Example invocation
99 :name: tab-tests-exec
100 :widths: auto
101
102 A, ``pismv -test A -Mx 61 -My 61 -Mz 11 -y 25000``
103 B, ``pismv -test B -Mx 61 -My 61 -Mz 11 -ys 422.45 -y 25000``
104 C, ``pismv -test C -Mx 61 -My 61 -Mz 11 -y 15208.0``
105 D, ``pismv -test D -Mx 61 -My 61 -Mz 11 -y 25000``
106 E, ``pismv -test E -Mx 61 -My 61 -Mz 11 -y 25000``
107 F, ``pismv -test F -Mx 61 -My 61 -Mz 61 -y 25000``
108 G, ``pismv -test G -Mx 61 -My 61 -Mz 61 -y 25000``
109 H, ``pismv -test H -Mx 61 -My 61 -Mz 11 -y 40034 -bed_def iso``
110 I, ``ssa_testi -ssa_method fd -Mx 5 -My 500 -ssafd_picard_rtol 1e-6 -ssafd_ksp_rtol 1e-11``
111 J, ``ssa_testj -ssa_method fd -Mx 60 -My 60 -ssafd_ksp_rtol 1e-12``
112 K, ``pismv -test K -Mx 6 -My 6 -Mz 401 -Mbz 101 -y 130000``
113 L, ``pismv -test L -Mx 61 -My 61 -Mz 31 -y 25000``
114
115 .. csv-table:: ``pismv`` command-line options
116 :header: Option, Description
117 :name: tab-pismv-options
118
119 ``-test``, Choose verification test by single character name; see :numref:`tab-tests`.
120 ``-no_report``, Do not report errors at the end of a verification run.
121 ``-eo``, Only evaluate the exact solution; no numerical approximation at all.
122 ``-report_file``, Save error report to a netCDF file.
123 ``-append``, Append to a report file.
124
125 :numref:`tab-tests` summarizes the many exact solutions currently available in PISM. Most
126 of these exact solutions are solutions of *free boundary problems* for partial
127 differential equations; only Tests A, E, J, K are fixed boundary value problems.
128
129 :numref:`tab-tests-exec` shows how to run each of them on a coarse grids. Note that tests
130 I and J require special executables ``ssa_testi,ssa_testj`` which are built with
131 configuration flag ``Pism_BUILD_EXTRA_EXECS`` equal to ``ON``. :numref:`tab-pismv-options`
132 gives the special verification-related options of the ``pismv`` executable.
133
134 Numerical errors are not, however, the dominant reasons why ice sheet models give
135 imperfect results. The largest sources of errors include those from using the wrong (e.g.
136 over-simplified or incorrectly-parameterized) continuum model, and from observational or
137 pre-processing errors present in input data. Our focus here on numerical errors has a
138 model-maintenance goal. It is *easier* to maintain code by quantitatively confirming that
139 it produces small errors in cases where those can be measured, rather than "eyeballing"
140 results to see that they are "right" according to human judgment.
141
142 The goal of verification is not generally to see that the error is zero at any particular
143 resolution, or even to show that the error is small in a predetermined absolute sense.
144 Rather the goals are
145
146 - to see that the error *is* decreasing,
147 - to measure the rate at which it decreases, and
148 - to develop a sense of the magnitude of numerical error before doing realistic ice sheet
149 model runs.
150
151 Knowing the error decay rate may give a prediction of how fine a grid is necessary to
152 achieve a desired smallness for the numerical error.
153
154 Therefore one must "go down" a grid refinement "path" and measure numerical error for each
155 grid :cite:`Roache`. The refinement path is defined by a sequence of spatial grid cell sizes
156 which decrease toward the refinement limit of zero size :cite:`MortonMayers`. In PISM the
157 timestep :math:`\Delta t` is determined adaptively by a stability criterion (see
158 section :ref:`sec-adapt`). In PISM one specifies the number of grid points, thus the
159 grid cell sizes because the overall dimensions of the computational box are normally
160 fixed; see section :ref:`sec-coords`. By "measuring the error for each grid" we mean
161 computing a norm (or norms) of the difference between the numerical solution and the exact
162 solution.
163
164 For a grid refinement path example, in tests of the thermomechanically-coupled SIA model
165 one refines in three dimensions, and these runs produced Figures 13, 14, and 15 of :cite:`BBL`:
166
167 .. code-block:: none
168
169 pismv -test G -max_dt 10.0 -y 25000 -Mx 61 -My 61 -Mz 61 -z_spacing equal
170 pismv -test G -max_dt 10.0 -y 25000 -Mx 91 -My 91 -Mz 91 -z_spacing equal
171 pismv -test G -max_dt 10.0 -y 25000 -Mx 121 -My 121 -Mz 121 -z_spacing equal
172 pismv -test G -max_dt 10.0 -y 25000 -Mx 181 -My 181 -Mz 181 -z_spacing equal
173 pismv -test G -max_dt 10.0 -y 25000 -Mx 241 -My 241 -Mz 241 -z_spacing equal
174 pismv -test G -max_dt 10.0 -y 25000 -Mx 361 -My 361 -Mz 361 -z_spacing equal
175
176 The last two runs require a supercomputer! In fact the :math:`361\times 361\times 361` run
177 involves more than :math:`100` million unknowns, updated at each of millions of time
178 steps. Appropriate use of parallelism (``mpiexec -n NN pismv``) and of the ``-skip``
179 modification to adaptive timestepping accelerates such fine-grid runs; see section
180 :ref:`sec-adapt`.
181
182 Figures :numref:`fig-thickerrsB` through :numref:`fig-velerrsI` in
183 :ref:`sec-convergence-plots` show a sampling of the results of verifying PISM using the
184 tests described above. These figures were produced automatically using Python scripts
185 ``test/vfnow.py`` and ``test/vnreport.py``. See section :ref:`sec-scripts`.
186
187 These figures *do not* show outstanding rates of convergence, relative to textbook partial
188 differential equation examples. For the errors in tests B and G, see the discussion of
189 free margin shape in :cite:`BLKCB`. For the errors in test I, the exact continuum solution is
190 not very smooth at the free boundary :cite:`SchoofStream`.
191
192 .. toctree::
193
194 convergence-figures.rst