Partial Differential Equations¶
Storing varios PDE’s that can be will be solved in this course. This includes:
Diffusive 1D
Advective 1D
Diffusive-Advective 1D
The goal is to implement the code in python and not rely on existing methods.
Bram Lagerweij COHMAS Mechanical Engineering KAUST 2021
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pde.advective(dof, dx, c)¶ Time derivative of the PDE for advective diffusive problems.
\[u_{t} + c u_{x} = 0 \qquad \forall \, x \in \Omega = [0, 1] \quad \& \quad t>0\]Thus this returns:
\[u_{t} = - c u_{x}\]Because we use finite difference based matrix products we can convert this into a matrix vector product, where \(D_x\) is the central difference approximation of \(\partial_x\):
\[u_{t} = -c D_{x} u = K u\]This function calculates the matrix \(K\). Because it should be compatible with general, non-homogeneous formulation, a part that is independent of \(u\) is also included.
- Parameters
dof (int) – Number of degrees of freedom.
dx (float) – Stepsize in the of spatial discretisation.
c (float) – The avective coefficient.
- Returns
K (matrix (sparse csr format)) – The time derivative part of the pde obtained from the spatial part.
b (vector (dense array)) – The remaining term, in this homeneous case it is a zero array.
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pde.advectivediffusive(dof, dx, mu, c)¶ Time derivative of the PDE for advective diffusive problems.
\[u_{t} + c u_{x} = \mu u_{xx} \qquad \forall \, x \in \Omega = [0, 1] \quad \& \quad t>0\]Thus this returns:
\[u_{t} = - c u_{x} + \mu u_{xx}\]Because we use finite difference based matrix products we can convert this into a matrix vector product, where \(D_x\) is the central difference approximation of \(\partial_x\) and similarly \(D_{xx}\) the central difference apprximation of \(\partial_{xx}\):
\[u_{t} = -c D_{x} u + \mu D_{xx} u = (-c D_{x} + \mu D_{xx})\, u = K u\]This function calculates the matrix \(K\). Because it should be compatible with general, non-homogeneous formulation, a part that is independent of \(u\) is also included.
- Parameters
dof (int) – Number of degrees of freedom.
dx (float) – Stepsize in the of spatial discretisation.
mu (float) – The diffusive coefficient.
c (float) – The avective coefficient.
- Returns
K (matrix (sparse csr format)) – The time derivative part of the pde obtained from the spatial part.
b (vector (dense array)) – The remaining term, in this homeneous case it is a zero array.
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pde.diffusive(dof, dx, mu)¶ Time derivative of the PDE for advective diffusive problems.
\[u_{t} = \mu u_{xx} \qquad \forall \, x \in \Omega = [0, 1] \quad \& \quad t>0\]Thus this returns:
\[u_{t} = \mu u_{xx}\]Because we use finite difference based matrix products we can convert this into a matrix vector product, where \(D_xx\) is the central difference approximation of \(\partial_{xx}\):
\[u_{t} = \mu D_{xx} u = K u\]This function calculates the matrix \(K\). Because it should be compatible with general, non-homogeneous formulation, a part that is independent of \(u\) is also included.
- Parameters
dof (int) – Number of degrees of freedom.
dx (float) – Stepsize in the of spatial discretisation.
mu (float) – The defusive coefficient.
- Returns
K (matrix (sparse csr format)) – The time derivative part of the pde obtained from the spatial part.
b (vector (dense array)) – The remaining term, in this homeneous case it is a zero array.