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version 0.6.0
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version 0.6.0
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Topics | |
| Schemes | |
| Scheme submodules. | |
Namespaces | |
| module | mod_finite_differences_computer |
| Computer for evaluating Finite Difference Schemes. | |
| module | mod_finite_differences_3d_stencils |
| Stencil filling for 3D schemes. | |
| module | mod_finite_differences_init |
| Initialization of declared finite differences schemes. | |
| module | type_fd_scheme |
| The Finite Difference Scheme type definition. | |
| module | type_fd_weno_scheme |
| The Finite Difference Weno Scheme type definition. | |
| module | type_discrete_stencil |
| The Finite Difference Scheme type definition. | |
This module permits to easily build and use finite difference schemes on the Notus grid or any discretized function. They can be used for implicit or explicit use. Various schemes are written for:
A finite difference scheme (noted fd_scheme) is defined as the list of coefficients (named weights and noted \(w_i\)) that corresponds to a discrete approximation of a certain derivative. It is based on the mathematical definition of the \(M^{th}\) order approximation of the \(n^{th}\) derivative of a function \(\phi\):
\[ \frac{d^n \phi}{d \xi^n} ( \xi_0 ) \simeq \sum_i w_i \phi( \xi_i ) + O(\Delta \xi^M) = \sum_i w_i \phi_i + O(\Delta \xi^M) \]
where \( \xi \) is any direction (being \(x\), \(y\) or \(z\)), \( \phi_i = \phi(\xi_i) \) are the discrete values of the function \(\phi\). By convention, we note \(\xi_0\) the position to which the finite difference scheme is related.
Each scheme is defined in a separate module/file with an explicit name, eg. the scheme first O2 for the first derivative ( \(d \phi / d \xi\)) with second order precision (O2). By convention, the associated functions are named fd_scheme_derivative_order_type, where:
first or secondO1 (order 1), O2 (order 2), etc.centered, forward or backwardSchemes can be directly used for implicit formulation or can be applied for explicit formulation (see below).
A scheme is defined as an array of real values. Each scheme is defined on a specific stencil (a mask on the grid, see the paragraph below) defined as effective elements surrounding a grid element. To each stencil element is associated a value called the weight. These values represent the factor (noted \(w_i\)) associated to each grid element, as presented in the main equation above. As we use a vector representation for weights, null values may appear inside the stencil.
The Fortran type is defined in type_fd_scheme.
The scheme stencil (ie. boundaries) is representend by the min and max indices outside of which the weights are all null. Thus, weights are only stored inside this stencil. The indices are relative to the element where the finite difference will be computed. The stencil's indices are stored in fd_scheme%index_start and fd_scheme%index_end. They are also specified in the documentation of each scheme.
All the instances of the schemes are (and has to be) created with the procedure fd_initialize()
Every schemes are built (returned) by the fd_scheme* functions.
The developer can define her/his own schemes. Please refer to the Schemes documentation for more information.
In an implicit formulation, one only needs the coefficients associated to each scheme. Thus, the developer will need to declare a variable of type fd_scheme and get the wanted scheme through the build functions detailed above. The coefficients are then accessible in the fd_scheme%weight array. Remember that the weights are indexed relatively to the incriminated discretization point.
my_scheme%weight(-2), my_scheme%weight(-1) and my_scheme%weight(0) the numerical values of the coefficients that would correspond to \(\phi_{-2}\), \(\phi_{-1}\) and \(\phi_0\), given that the steps between those points are \(0.1\) and \(0.2\).An explicit formulation requires that the developer evaluates the finite difference through given values (usualy the values taken from the grid). To do so, there are two ways:
In this section we furnish a complete Fortran example for numericaly solving the 1D heat equation.
Consider the following heat equation:
\[ \frac{\partial \phi}{\partial t} - \mu \frac{\partial^2 \phi}{\partial x^2} = 0 \]
For the formulations below, we assume that the temporal derivative is discretized with a second order backward scheme:
\[ \frac{\partial \phi}{\partial t} \simeq \alpha \phi^{n+1}_i + \beta \phi^{n}_i + \gamma \phi^{n-1}_i \]
with possible variable time step and where superscripts denotes temporal indices and subscripts denotes spatial indices.
The following code is used as a preamble for both implicit and explicit formulation, defining the known values:
where the wanted result will be stored in phi_np1.
The spatial discretization is treated implicitely with a second order centered scheme (assuming that the step is uniform, for simplicity) as:
\[ \frac{\partial^2 \phi}{\partial x^2} \simeq = \frac{\phi^{n+1}_{i-1} - 2 \phi^{n+1}_{i} + \phi^{n+1}_{i+1}}{\Delta x} \]
The complete numerical scheme reads:
\[ \alpha \phi^{n+1}_i - \mu \frac{\phi^{n+1}_{i-1} - 2 \phi^{n+1}_{i} + \phi^{n+1}_{i+1}}{\Delta x} = - \beta \phi^n_i - \gamma \phi^{n-1}_i \]
with the implicit part on the left-hand side of the equation and the explicit part on the right-hand side. The corresponding Fortran code would be:
where the particular treatment needed for the first time step and the spatial boundaries haven't been detailed.
The explicit formulation is quite similar to the implicit one presented above. The spatial discretization is treated explicitely with a second order centered scheme (assuming that the step is uniform, for simplicity) as:
\[ \frac{\partial^2 \phi}{\partial x^2} \simeq = \frac{\phi^{n}_{i-1} - 2 \phi^{n}_{i} + \phi^{n}_{i+1}}{\Delta x} \]
The complete numerical scheme reads:
\[ \phi^{n+1}_i = \alpha^{-1} \left( - \beta \phi^n_i - \gamma \phi^{n-1}_i + \mu \frac{\phi^{n}_{i-1} - 2 \phi^{n}_{i} + \phi^{n}_{i+1}}{\Delta x} \right) \]
The corresponding Fortran code would be:
where the particular treatment needed for the first time step and the spatial boundaries haven't been detailed.
1.9.5