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The Method of Manufactured Solutions (MMS) Verification Driver

Input

The MMS driver accepts input in YAML form, like the main RDycore driver. However, the MMS driver's input has a Ñ•lightly different form. Like the main driver input, it's organized into several sections. Many of these sections are identical to those in the main driver's input:

However, the other sections, which define material properties, initial/boundary conditions, and sources, are not present in the MMS input. This is because the method manufactured solutions requires analytic forms for these terms to produce a convergent manufactured solution. So these sections are replaced by a single mms section that defines these analytic forms.

mms section

mms:
  constants: # any single capital letter can be used
    H: 0.005  # water height scale factor
    T: 20.0   # time scale
    U: 0.025  # x-velocity scale factor
    V: 0.025  # y-velocity scale factor
    N: 0.01   # manning coefficient scale factor
    Z: 0.0025 # elevation scale factor

    K: 0.6283185307179586 # wave number in x and y (pi/5)
  swe: # functions of x, y, t (non-normalized units)
    # water height
    h:    H * (1 + sin(K*x)*sin(K*y)) * exp(t/T)
    dhdx: H * K * sin(K*y) * cos(K*x) * exp(t/T)
    dhdy: H * K * sin(K*x) * cos(K*y) * exp(t/T)
    dhdt: H / T * (1 + sin(K*x)*sin(K*y)) * exp(t/T)

    # x velocity
    u:     U * cos(K*x) * sin(K*y) * exp(t/T)
    dudx: -U * K * sin(K*x) * sin(K*y) * exp(t/T)
    dudy:  U * K * cos(K*x) * cos(K*y) * exp(t/T)
    dudt:  U / T * cos(K*x) * sin(K*y) * exp(t/T)

    # y velocity
    v:     V * sin(K*x) * cos(K*y) * exp(t/T)
    dvdx:  K * V * cos(K*x) * cos(K*y) * exp(t/T)
    dvdy: -K * V * sin(K*x) * sin(K*y) * exp(t/T)
    dvdt:  V / T * sin(K*x) * cos(K*y) * exp(t/T)

    # elevation as z(x, y)
    z:     Z * sin(K*x) * sin(K*y)
    dzdx:  Z * K * cos(K*x) * sin(K*y)
    dzdy:  Z * K * sin(K*x) * cos(K*y)

    # Manning coefficient n(x,y)
    n:     N * (1 + sin(K*x) * sin(K*y))

  # Convergence study parameters (optional)
  convergence:
    num_refinements: 3
    base_refinement: 1
    expected_rates:
      h:
        L1: 1
        L2: 1
        Linf: 0.48
      hu:
        L1: 0.73
        L2: 0.78
        Linf: 0.62
      hv:
        L1: 0.73
        L2: 0.78
        Linf: 0.62

The mms section defines the forms of the manufactured solutions for the model equations corresponding to parameters set in the physics section.

The constants subsection defineѕ a set of named constants that can be used in the analytic forms for the manufactured solutions. Any single capital roman letter (A through Z) can be used as a constant. In the above example, the solutions reference the constants H, T, U, V, N, Z, and K, which are defined as shown.

The swe subsection defines a set of manufactured solutions to the 2D shallow water equations (SWE) in terms of a water height h with a flow velocity (u, v). Each of the components h, u, v are represented by a function of the coordinates x and y and the time t. Other model parameters (z, the elevation function, and n, the Manning coefficient) are functions of x and y only.

These analytic forms are parsed and compiled at runtime so they can be evaluated as needed by the model. This means you can define a new manufactured solution in every MMS driver input file, without developing code and rebuilding RDycore.

Convergence studies

The optional convergence subsection contains the following parameters for performing convergence studies that determine whether the MMS problem has been solved successfully for each solution component:

  • num_refinements: the number of times the domain (and timestep) are refined uniformly from the base resolution to test the rate of convergence of the solution error. This parameter is required.
  • base_refinement: this optional parameter specifies the number of times the mesh should be refined to establish the coarsest resolution to be used in the convergence study. For example, a base_refinement of 2 indicates that a mesh loaded from a file should be refined twice before performing a convergence study.
  • expected_rates: a sub-subsection with L1, L2, and Linf entries for each relevant solution component name giving the expected rates of convergence for the appropriate error norms. Each of the component names and expected rates are optional, so you can specify only those you want to use as pass/fail criteria.

NOTE: When the MMS driver performs a convergence study, it writes no output. If you need to write a mesh or solution data, you can always use an input file without the convergence section to compute error norms for a single spatial resolution, writing output as needed.