heatcompact3d: A Fortran repository from tlestang

Prototype for 3D heat equation solver

$$\frac{\partial T}{\partial t}(x, y ,z) = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}$$

Example

use field, only: field_type
use time_integration: AB2_integrator_type

temp_field = field_type(u0, dx)
AB2 = AB2_integrator_type(starttime=0., endtime=0.3, dt)

call AB2%integrate(temp_field)

temp_field%dump("output.dat")

Building and run the tests

Configure with

cmake -S . -B build

Tests are built and run from the build directory:

make && make test

Generating and viewing the documentation

The documentation is generated using ford. It is deployed automatically from the main branch at https://tlestang.github.io/heatcompact3d/.

To preview the docs locally, you'll need ford:

pip install --user ford

Then, from the root of the repository:

ford heatcompact3d.md

You can then view the documentation in your browser, for instance:

firefox doc/index.html

Layout

High-level structure

Two client facing derived-types: integrator_type and field_type.

Derived type field_type (module_field.f90) models a three-dimensional temperature field satisfying the heat equation discretized over a unifomr spatial mesh of spacing dx. Type-bound function rhs of field_type returns a new field_type instance evaluating the right-hand side of the heat equation ($\Delta T$) over the mesh.

Type time_integrator_type (module_time_integrator.f90) provides a base abstract derived type for time integration methods such as Euler, Adams-Bashforth or Runge-Kutta methods. Type-bound method integrate of time_integrator_type takes a field_type instance and overwrites it with the result of the time integration. Currently available integrators:

euler_integrator_type
AB2_integrator_type
RK3_integrator_type

Spatial differentiation

Spatial differentation for both first order and second order derivatives is implemented in the differentiate module (module_differentiate.f90). The differentiate module provides access to derived type differentiator_type, which diff type-bound function implements differentiation of a one dimensional pencil, /e.g/

$$g(i) = af_{i-2} + bf_{i-1} + af_{i+1} + bf_{i+2}$$

Instances of differentiator_type are created through constructors <method>_<order>. Currently available differentiatiors are:

sixth_order_compact_{1,2}

Differentiation stencils

A differentiation stencil of size $m$ takes the form

$$ug_{i+1} + g + lg_{i-1} = \sum_{k=1}^{m} w_k f_{i+s_k}$$

it is modeled by derived-type stencil_type with the following components

nodes (${s_k}_{1 \ leq k \leq m}$) :: stencil node's position relative to point to which the stencil is applied.
coeffs (${w_k}_{1 \ leq k \leq m}$)
upper, lower ($u$, $l$) coefficients forming the lower and upper diagonals of the tridiagnonal matrix describing the system.

A instance of stencil_type can be applied along a one dimensional pencil

s = stencil_type(...)
f, rhs = real(*)
rhs = s%apply_along(f)

Boundary schemes

Module boundary (boundary_mod.f90) composes four instances of stencil_type.

East boundary stencil (first order derivative)
West boundary stencil (first order derivative)
East boundary stencil (second order derivative)
West boundary stencil (second order derivative)

The boundary provides access to specific boundary conditions through special-purpose constructors. Currently implemented boundary schemes are

get_dirichlet_boundary :: Returns a boundary_type instance describing a dirichlet boundary condition.

tlestang/heatcompact3d