solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis Class Reference

The Gauss-Legendre Basis is by construction the only basis which yields diagonal mass matrices. More...

Inheritance diagram for solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis:

Collaboration diagram for solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis:

Public Member Functions
	__init__ (self, polynomial_order)
	quadrature_points (self)
	It seems to be a little bit weird that I have to use the polynomial order plus one here, but you have to read the docu to understand that we are evaluating the polynomial of a certain order to get the roots.
	quadrature_weights (self)
	It seems to be a little bit weird that I have to use the polynomial order plus one here, but you have to read the docu to understand that we are evaluating the polynomial of a certain order to get the roots.
Public Member Functions inherited from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix
	__str__ (self)
	init_dictionary_with_default_parameters (self, dictionary, use_multidimensional_arrays)
	To be implemented by subclass.
	value1d (self, x, number)
	Return the numberth shape function's value at x.
	derivative1d (self, x, number)
	Return the numberth shape function's derivative at x.
Public Member Functions inherited from solvers.LagrangeBasis.LagrangeBasis
	order (self)
	dofs_per_axis (self)

Additional Inherited Members
Data Fields inherited from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix
	mass_matrix
	dofs_per_axis
Static Public Attributes inherited from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix
str	QuadraturePoints1dDocumentation
str	QuadratureWeights1dDocumentation
str	MassMatrix1dDocumentation
str	StiffnessOperator1dDocumentation
str	BasisFunctionValuesLeft1dDocumentation
str	DerivativeOperator1dDocumentation
str	InterpolationMatrix1dDocumentation
str	RestrictionMatrix1dDocumentation
	stiffness_matrix = None
Protected Attributes inherited from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix
	_polynomial_order
Protected Attributes inherited from solvers.LagrangeBasis.LagrangeBasis
	_polynomial_order
	_render_digits

Detailed Description

The Gauss-Legendre Basis is by construction the only basis which yields diagonal mass matrices.

I rely solely on SciPy here to construct these functions, as I'm interested in numerical values only, and as the normal SciPy accuracy is sufficient.

There might be more elegant implementations using SymPy which exploit the special structure behind Gauss-Legendre. But I prefer here to stick to one Python package only.

Definition at line 615 of file LagrangeBasisWithDiagonalMassMatrix.py.

Constructor & Destructor Documentation

__init__()

solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis.__init__	(		self,
			polynomial_order
	)

Reimplemented from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.

Definition at line 629 of file LagrangeBasisWithDiagonalMassMatrix.py.

References solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis.__init__().

Referenced by solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis.__init__().

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Member Function Documentation

quadrature_points()

solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis.quadrature_points

(

self

)

It seems to be a little bit weird that I have to use the polynomial order plus one here, but you have to read the docu to understand that we are evaluating the polynomial of a certain order to get the roots.

So the argument has to be equal to the number of quadrature points that I wannt have.

I originally used the roots_legendre and then did recalibrate the result manually. Today, SciPy directly offers a shifted Legendre function which I can use directly. So I don't need an expression like

return [ x/2.0+0.5 for x in roots_on_minus_one_one_interval ]

anymore. The nicest definition of the shifted Legendre shape functions can, as so often, be found on Wikipedia:

https://en.wikipedia.org/wiki/Legendre_polynomials#Shifted_Legendre_polynomials

Reimplemented from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.

Definition at line 634 of file LagrangeBasisWithDiagonalMassMatrix.py.

References solvers.LagrangeBasis.LagrangeBasis._polynomial_order, and solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix._polynomial_order.

Referenced by solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.__compute_derivative_operator(), solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.__compute_restriction_matrix(), solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.derivative1d(), solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.init_dictionary_with_default_parameters(), and solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.value1d().

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quadrature_weights()

solvers.LagrangeBasisWithDiagonalMassMatrix.GaussLegendreBasis.quadrature_weights

(

self

)

It seems to be a little bit weird that I have to use the polynomial order plus one here, but you have to read the docu to understand that we are evaluating the polynomial of a certain order to get the roots.

So the argument has to be equal to the number of quadrature points that I wannt have.

I originally used the roots_legendre and then did recalibrate the result manually. Today, SciPy directly offers a shifted Legendre function which I can use directly. So I don't need an expression like

return [ x/2.0+0.5 for x in roots_on_minus_one_one_interval ]

anymore. The nicest definition of the shifted Legendre shape functions can, as so often, be found on Wikipedia:

https://en.wikipedia.org/wiki/Legendre_polynomials#Shifted_Legendre_polynomials

Reimplemented from solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.

Definition at line 662 of file LagrangeBasisWithDiagonalMassMatrix.py.

References solvers.LagrangeBasis.LagrangeBasis._polynomial_order, and solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix._polynomial_order.

Referenced by solvers.LagrangeBasisWithDiagonalMassMatrix.LagrangeBasisWithDiagonalMassMatrix.init_dictionary_with_default_parameters().

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The documentation for this class was generated from the following file:

• python/exahype2/solvers/LagrangeBasisWithDiagonalMassMatrix.py