model_problems - apusok/FD-PDE GitHub Wiki

Model Problems

Below we summarize a series of model problems which will be useful in testing / verifying code implementations required for simulating two-phase rift dynamics. Model problems are catergorised by their PDE structure (cf. catergorisation by physical application)

Model problems are identified of as type "manufactured", "analytic", or "benchmark".

  1. For all "Manufactured" tests, the primary unknowns of the PDE can be directly computed with the chosen solution. These tests can be used to measure discretization error and thus can be used to measure the order of accuracy of the discretization. "Manufactured" tests can be used for fully automated regression testing.
  2. "Analytic" tests may (or may not) provide definitions of the primary unknowns of the PDE. Hence they may be able to be used for measuring the order of accuracy of the discretization and regression testing.
  3. "Benchmark" tests provided tabulations of quantities of interest (metrics) derived from the primary variables. Comparisons involving "benchmark" tests will generally require one to compare the output metrics with previous published values. Cannot be used for measuring the order of accuracy, or for reliable automated regression testing.

Single phase flow

Instantaneous Rift (type: benchmark)
  • Key "may-rift-demo"
  • Description
    • Rectangular domain [-1,1] x [0,1]
    • Boundary conditions
      • Extensional: u.n = +U (x=1), u.n = -U (x=-1). t.tau.n = 0 (x=1, x=-1)
      • Basal boundary condition (y=0): u.n = 0, t.tau.n=0
      • Surface boundary condition (y=1): t.sigma.n = n.sigma.n = 0
    • Three regions defined: rock, inclusion, air
      • Rock: domain = [-1,1] x [0,0.9], rheology = visco-plastic
      • Air: [-1,1] x [0.9,1], rheology = viscous (very weak), density = 0
      • Inclusion: [-0.1,0.1] x [0,0.1], rheology = viscous (very strong or very weak)
  • Solution characteristics
    • "Toy rift with Instantaneous shear zone development"
    • Time independent.
    • Visco-plastic (strain-rate dependent J2 flow law).
    • Purely mechanical - no thermal (temperature) dependence.
    • Verifies FD-PDE is flexible.
    • Inspired by the brick models, such as that defined here
      • Spiegelman, M., May, D.A. and Wilson, C.R., 2016. On the solvability of incompressible Stokes with viscoplastic rheologies in geodynamics. Geochemistry, Geophysics, Geosystems, 17(6), pp.2213-2238.
  • Comments
    • The model requires some approximation of a free surface. This can implemented in numerous ways
      • Sticky-air mode 1: Define air layer to be thick (say 33% of entire domain); Set free-slip boundary conditions on upper surface; Impose exentsional boundary conditions on left/right sides throughout the air region.
      • Sticky-air mode 2 (default): Define air layer to be thin (say 10% of entire domain); Set free-surface boundary conditions on upper surface; Impose exentsional boundary conditions on left/right sides throughout the air region.
      • Use method of Duretz et al (it's complicated to implement).
        • Duretz, T., May, D.A. and Yamato, P., 2016. A free surface capturing discretization for the staggered grid finite difference scheme. Geophysical Journal International, 204(3), pp.1518-1530.

Two phase flow

Rhebergen - case 2 (type: manufactured)
  • Key "rhebergen-siam-2015"
  • Reference
    • Rhebergen, S., Wells, G.N., Wathen, A.J. and Katz, R.F., 2015. Three-field block preconditioners for models of coupled magma/mantle dynamics. SIAM Journal on Scientific Computing, 37(5), pp.A2270-A2294..
    • Problem defined in section 6.2 - see equations (6.1) - (6.4)
    • Online source: https://arxiv.org/pdf/1411.5235.pdf
  • Solution characteristics
    • Two-dimensional, defined over the domain [0,1] x [0,1].
    • Time independent.
    • Spatially varying porosity. Upper and lower bounds are specified as input.
    • Porosity dependent shear and bulk viscosity with realistic flow laws.
    • Porosity depdent permeability with realistic functional form.
    • Sin / Cosine velocity and pressure solutions.
Katz - case 2 (type: benchmark)
  • Key "katz-jcp-2008-case-2"
  • Reference
    • Katz, R.F. and Worster, M.G., 2008. Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell. Journal of Computational Physics, 227(23), pp.9823-9840.
    • Problem defined in section 3.2
    • Additional references which define this problem, and code results from other codes is provided in the above reference.
  • Solution characteristics
    • "Convection in a fixed porous medium".
    • Two-dimensional.
    • No phase change. Hence this can be thought of as a analog of the single phase convection benchmark.
    • Solution accuracy is inferred from comparisons with other implementations / software.
    • Quantity of interest used in comparison is the mean Nusselt number.

Enthalpy Method

Krabbenhoft (type: analytic)
  • Key "krabbenhoft-h&mtrans-2007"
  • Reference
    • Krabbenhoft, K., Damkilde, L. and Nazem, M., 2007. An implicit mixed enthalpy–temperature method for phase-change problems. Heat and mass transfer, 43(3), pp.233-241.
    • Problem defined in section 5.1
  • Solution characteristics
    • "One-dimensional solidification".
    • 1D analytic solution for temperature.
    • Phase diagram (H-T) is linear, hence closed form expressions for T (given H) exists.
    • Problem only involves diffusion (no advection).
Feulvarch (type: analytic)
  • Key "feulvarch-numerheattrans-2007"
  • Reference
    • Feulvarch, E. and Bergheau, J.M., 2007. An implicit fixed-grid method for the finite-element analysis of heat transfer involving phase changes. Numerical Heat Transfer, Part B: Fundamentals, 51(6), pp.585-610.
    • Problem defined in section 3.3
  • Solution characteristics
    • "Isothermal transformation".
    • 1D analytic solution for temperature.
    • Phase diagram (H-T) is linear, hence closed form expressions for T (given H) exists.
    • Problem involves both diffusion and advection.
Katz - case 1 (type: analytic)
  • Key "katz-jcp-2008-case-1"
  • Reference
    • Katz, R.F. and Worster, M.G., 2008. Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell. Journal of Computational Physics, 227(23), pp.9823-9840.
    • Problem defined in section 3.1
  • Solution characteristics
    • "Solidification without fluid flow".
    • Two-dimensional.
    • Semi infinite domain.
    • Steady state solution.
    • Heat capactity and thermal conductivity assumed constant.
    • 1D temperature profiles are used for verification.
    • Phase diagram requires a (local) nonlinear solve. Phase diagram contains a eutectic point.

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