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

2. 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

3. Structured Grids versus Unstructured Grids Spatial Discretisation
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

4. Spatial Discretisation
Finite Element Discretisation:
Reference element
Physical element
Physical space Reference space
Physical element Reference element
Function approximation
Integral method Integration by parts
Weighted residuals Galerkin method
PDE discretisation method
Numerical integration Gauss method
Numerical integration
Spatial discretisation-Summary

5. Basic Properties Truncation error Consistency Stability
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
Lax’s 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

6. 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

7. 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

8. 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

9. Taylor Galerkin Schemes
The Taylor-Galerkin schemes may be considered as a generalisation of the explicit Euler scheme (Padé
polynomals with q=0):
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

10. 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

11. 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

12. 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

13. 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

14. Numerical Schemes Property
Discretisation in time

15. 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

16. 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

17. 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

18. Numerical Schemes Property
Discretisation
Time
Space FD
Time
continuous FE FD
Euler
scheme FE LW
LW-FE
LW-TG
Discretisation in time

19. Numerical Schemes Property
Discretisation in time

20. 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

21. 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

22. 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

23. 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

24. Extension to multi dimensional problems
Multidimensional discretisation property x y
Domain of numerical stability of the LW schemes
Phase velocity error of the LW schemes
Discretisation in time

25. 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

26. Convection of a product cosine hill
LW-FE(Md) LW
LW-FE
LW-TG
Convection of a hill in a rotational field, ∆t=2π/200, after one complete rotation
LW-FE(Md) LW
LW-FE: unstable
LW-TG
Convection of a hill in a rotational field, ∆t=2π/120, after one complete rotation
Discretisation in time

27. Convection of a product cosine hill
LW-TG
Convection of a hill in a rotational field, ∆t=2π/200, after 3/4 and 1 complete rotation
Discretisation in time

28. 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

29. 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
TTG=LW-TG2
Discretisation in time

30. Two step Taylor Galerkin Schemes
Propagation of a cosine profile
Discretisation in time

31. 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

32. 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.
Second order:
Third order:
Fourth order:
Maximum stability condition:
Discretisation in time

33. 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 l+1 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