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Time Discretisation - Taylor-Galerkin Schemes
V. Selmin
Multidisciplinary Computation and Numerical
Simulation
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Outline

• Spatial discretisation summary
• Basic properties of numerical schemes
• Time discretisation
• Taylor-Galerkin schemes
• - Basic Taylor-Galerkin schemes
• - Extension to non-linear problems
• - Extension to multi-dimensional problems
• - Two-steps Taylor-Galerkin schemes
• Multi-stages algorithms

Outline
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Spatial Discretisation
Structured Grids versus Unstructured Grids
Structured grids Same number of cells around a
node
Unstructured grids The number of cells around a
node is not the same
Spatial Discretisation
Finite Difference Discretisation
Taylor-series expansion
Finite Volume Discretisation
Integral formulation Divergence theorem
Spatial discretisation-Summary
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Spatial Discretisation
Finite Element Discretisation
Spatial discretisation-Summary
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Basic Properties
Truncation error Difference between the original
partial differential equation (PDE) and the
discretised equation (DE).

Consistency Consistency deals with the
extent to which the discretised equations
approximate the partial differential equations.
A discretised representation of the PDE is said
to be consistent if it can be shown that the
difference between the PDE and its discretised
representation vanishes as the mesh is
refined Stability Numerical stability is a
concept applicable in a strict sense only to
marching problems. A stable numerical scheme is
one for which errors for any source (round-off,
truncation, ) are not permitted to grow in the
sequence of numerical procedures as the
calculation proceeds from one marching step to
the next. Convergence of Marching
Problems Laxs Equivalence Theorem Given a
properly posed initial value problem and a
discretised approximation to it that satisfies
the consistency conditions, stability is
necessary and sufficient condition for
convergence.
Basic properties
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Discretisation in Time
Model equation Unsteady/steady problems If
the solution u is steady, u solution of is
also solution of the following pseudo-unsteady
problem Finite Difference 1- FD approximation
of the time derivative Spatial discretisation ?
Time discretisation ?
Steady Euler equations elliptic for subsonic
flows hyperbolic for supersonic flows
Discretisation in time
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Taylor series expansion
2- Taylor-series expansion Taylor-series
expansion of ? Replace the time
derivatives by using the equation That
leads to the following equation which has to
be discretised in space
Discretisation in time
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Padé Polynomials
The equality may be rewritten in the more
concise form A family of temporal schemes may be
buit by using the Padé polynomials approximation
of the exponential function. It consists to
approximate the function H(v) by the ratio of
two polynomials of order p and q, respectively,
with an error of
Explicit schemes
Implicit schemes
Discretisation in time
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Taylor Galerkin Schemes
The Taylor-Galerkin schemes may be considered as
a generalisation of the explicit Euler scheme
(Padé polynomals with q0) The time
derivatives are replaced by the expressions
obtained by using successive differentiation of
the original equation The third order
derivative is expressed in terms of a mixed
space-time form in order to allow the use of
finite element for the spatial
discretisation. In this term the time derivative
is replaced by a finite difference approximation
that maintains the global troncature
error The time discretised equation is
written according to where
Discretisation in time
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Taylor Galerkin Schemes
If the convention
is adopted for the scalar product on the
computational domain, the Galerkin equation at
node j corresponds to Explicitley, we got after
integration by parts of the second derivatives
terms In the case of piecewise linear
shape function, ETG2 and ETG3 schemes take the
form where is the Courant
number,
Discretisation in time
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Taylor Galerkin Schemes
In the right-hand side of the discretised
equation we may recognize the same term as the
Law-Wendroff scheme In addition, in the
left-hand side of those equations, we may
regognize the classical consistent mass of the
finite element theory which corrisponds to to
the operator . In the TG3 scheme,
it is modified by the additional term that
appears in the time discretised equation.
Remarks Due to the coupling terms, the presence
of the mass matrix represent a disadvantage from
the point of view of the computational time.
Nevertheless, it is possible to exploit its
effect in an explicit context. The following
iterative procedure may be used where
Discretisation in time
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Numerical Schemes Property
1- Von Neuman analysis method The Von Neumann
procedure consits in replacing each term
of the discretised equation by the
Fourier component of order k of an harmonic
decomposition of where is the
Fourier component of order k. The amplification
factor G is defined by the equality In
general, it is a complex number which may be
written on the following form where and
are respectively the module and the phase
of G . The stability condition of von Neuman
states that, for each Fourier mode, the
amplification factor must have a module limited
by a quantity enough close to unity for all value
of and . The explicit expression of
this criteria is The term
emphasizes that in some physical process, the
modes may increase exponentially and this
divergence does not be confused with an
unstability of the numerical method
Discretisation in time
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Numerical Schemes Property
For the previous numerical schemes, the
amplification factor takes the form where
and is a real
number The stability condition for the
three schemes is The reduction of
stability for TG2 is due to the consistent mass
matrix . The correction contained in the TG3
scheme allows to recover the stability
condition and the unit CFL property
that states that the signal propagates without
distorsions when .
Discretisation in time
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Numerical Schemes Property
Discretisation in time
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Numerical Schemes Property
In the case the spatial discretisation is
performed by maintaining the time continuous, the
following schemes are obtained for the
finite differences, and for piecewise linear
elements The
consistent mass matrix is responsable of the
better acurracy on the phase.
Discretisation in time
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Numerical Schemes Property
2- Modified equation method The Modified
Equation method consists a- To perform a Taylor
series expansion about of all the terms
of the discretised equation. b- To replace all
the time derivatives of order greater to one and
the mixted space-time by using the equation
obtained at the previous step Following this
procedure, we obtain the partial differential
equation of infinite order genuinely solved by
the numerical scheme The modified equation may
be written according to where the are
real coefficients. Let consider a elementary
solution where k is real and
is a complex number, the and
have to satisfy the following relations
Discretisation in time
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Numerical Schemes Property
In the limit case where (large
wave lenghts), we can negelect all the terms
except the non-zero coefficients of the lowest
order which will be denoted by r . In this
case The necessary stability condition
becomes In addition,
Discretisation in time
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Numerical Schemes Property
Discretisation
Time
Space
FD
Time continuous
FE
FD
Euler scheme
FE
LW
LW-FE
LW-TG
Discretisation in time
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Numerical Schemes Property
Discretisation in time
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Propagation of a cosine profile
To illustrate and compare the performance of the
schemes discussed so far, consider the advection
problem over the interval 0,1 and defined by
the following initial and boundary conditions
LW
LW-FE
LW-TG
Discretisation in time
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Extension to non linear convection
Let consider the following hyperbolic
equation Written in the quasi-linear
form it may be interpreted as a non linear
convection equation for which each point of the
solution propagates with a different
velocity. As in the previous case, the equation
is discretised in time by using the series
expansion in which the time derivatives are
replaced by using the original equation and its
successive differentiation
Discretisation in time
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Extension to non linear convection
By using the following identities the
expression of the third derivative in time is
equivalent to the following form Then, in the
nonlinear case, the equation discretised in time
may be written according to where
The consistent mass matrix depends of the unkown
Remarks In the case of a scalar equation (and
only) the third order time derivative may be
written in the following compact form
Discretisation in time
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Extension to multi dimensional problems
Let consider the following hyperbolic
equation The time derivatives may be expressed
according to By using the following
identities the expression of the third
derivative in time is equivalent to the following
form Then, in the case of multidimensional
problems, the equation discretised in time may be
written according to where
Discretisation in time
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Extension to multi dimensional problems
Multidimensional discretisation property
Domain of numerical stability of the LW schemes
Phase velocity error of the LW schemes
Discretisation in time
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Convection of a product cosine hill
To illustrate and compare the performance of the
schemes discussed so far, the advection of a
product cosine hill in a pure rotation velocity
field is considered. The initial condition
is The unknown has to be prescribed on the
inflow boundary and leave free on the outflow
boundary
Discretisation in time
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Convection of a product cosine hill
Convection of a hill in a rotational field,
?t2p/200, after one complete rotation
Convection of a hill in a rotational field,
?t2p/120, after one complete rotation
Discretisation in time
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Convection of a product cosine hill
Convection of a hill in a rotational field,
?t2p/200, after 3/4 and 1 complete rotation
Discretisation in time
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Two step Taylor Galerkin Schemes
A third-order T-G scheme which can be effectively
employed in non-linear multidimensional advection
problems is obtained by considering the
following two step procedure where the
value of the parameter is left unspecified
for the time being. In fact while the other
coefficients in the two equations assume the
value necessary for a third-order accuracy of the
two discretisations combined, the parameter
enters only the coefficient of fourth-order term
in the overall series and therefore its value
affects the amplification factor only within the
fourth-order accuracy. The fully discrete
counterpart of the equations above in the case of
linear advection problem in one dimension
is The value of the parameter may be
fixed in such a way to reproduce the excellent
phase accuracy of the LW-TG scheme
Discretisation in time
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Two step Taylor Galerkin Schemes
The modified equation is The third order
accuracy is preserved by the two-step procedure.
Moreover, the new scheme has the same phase
response. The amplification factor takes the
form
Property analysis
TTGLW-TG2
Discretisation in time
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Two step Taylor Galerkin Schemes
Propagation of a cosine profile
Discretisation in time
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Two step Taylor Galerkin Schemes
Multidimensional discretisation property
LW-TG
LW-TG2
Domain of numerical stability of the LW schemes
Amplification factor of the LW schemes
Discretisation in time
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Multiple stages algorithms
Another important family of time integration
schemes that account for only two time levels
are those named multi-stage algorithms
(generalisation of the Runge-Kutta method). Let
write the spatial discretised scheme as
follows where is the residual of
the partial differential equations The general
form of an explicit multi-stage algorithm with l
levels may be written according
to where
and for consistency.
Discretisation in time
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Hybrid Multi stages algorithms
In the case of the solution of steady problems,
the objective is to have an efficient integration
method which allows to get the steady state as
fast as possible while the time accuracy is
unimportant. That leads to a class of hybrid
schemes of the multi-stage tipe where the
residual of a generic l1 stage is evaluated
according to where and
are the convective and dissipative terms,
respectively. In order to satisfy the consistency
property, the following relations have to be
satisfied
Example
Discretisation in time