joshualampert / dispersiveshallowwater.jl Goto Github PK

I found it helpful to return NamedTuples instead of separating values and names. For example, you could return a NamedTuple from integrals(analysis_callback), e.g.,

integrals(analysis_callback) = (waterheight_total = first_vector, velocity = second_vector, entropy=  third_vector)

Then, you can iterate over the pairs(integrals(analysis_callback)) for plotting

It's just an idea - your current setup works fine of course

In Summation by Parts Operators for Finite Difference Approximations of Second-Derivatives with Variable Coefficients (2012) by K. Mattsson, SBP operators for second derivatives with variable coefficients have been introduced. These could be used here to use narrow-stencil SBP operators for the semidiscretization for the BBM-BBM equations with variable bathymetry. In SummationByPartsOperators.jl they are accessible by var_coef_derivative_operator and source Mattsson2012().

EDIT: SummationByPartsOperators.jl does not implement variable coefficient second-derivative SBP operators for periodic boundary conditions yet.

EDIT2: See ranocha/SummationByPartsOperators.jl#258

To date, the RelaxationCallback only conserves the given quantity. It is also possible to preserve dissipative semidiscretizations via relaxation.

This will allocate new vectors whenever it's called. That's of course totally fine for now but may slow down the process a bit when you use bigger state vectors (increased grid refinement).

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It would be nice to have a SummaryCallback similarly to Trixi.jl that uses TimerOutputs.jl to check some timings and allocations.

There are still some parts that are not included in the tests.

I just wonder a bit why you use RK4(). It's of course fine since many people would pick it as default with fixed step size, but at least for adaptive time stepping some thing like Tsit5() should often be more efficient.

It would be nice to support other dispersive wave equations. These include especially the equations in H. Ranocha, D. Mitsotakis, D. Ketcheson, A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations (2021) and the corresponding discretizations using SBP operators developed there. These include the following scalar equations with constant bottom:

• BBM
• Fornberg-Whitham
• Camassa-Holm
• Degasperis-Procesi
• Holm-Hone

Of course the classical

• Korteweg-de Vries (KdV) equation

would be of interest, too. Implementing these semidiscretizations should be pretty straightforward within DispersiveShallowWater.jl. Other interesting models to look at (requiring to develop energy preserving semidiscretizations using SBP operators first) would be the

• Serre-Green-Naghdi equations (e.g. Busto et al., On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems (2021))
• equations by Escalante and Morales de Luna (Escalante and Morales de Luna, A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation (2020))