Text of Time Discretisation - Taylor-Galerkin Schemes V. Selmin Multidisciplinary Computation and Numerical...
Time Discretisation - Taylor-Galerkin Schemes V. Selmin Multidisciplinary Computation and Numerical Simulation
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
Spatial Discretisation Spatial discretisation-Summary 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: Finite Volume Discretisation: Taylor-series expansion Integral formulation Divergence theorem
Spatial Discretisation Spatial discretisation-Summary Reference elementPhysical element Finite Element Discretisation: 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
Basic properties 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.
Discretisation in Time 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 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 Taylor series expansion
Discretisation in time The equality may be rewritten in the more concise form A family of temporal schemes may be buit by using the Pad 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 Pad Polynomials Explicit schemes Implicit schemes
Discretisation in time Taylor Galerkin Schemes The Taylor-Galerkin schemes may be considered as a generalisation of the explicit Euler scheme (Pad 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 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 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 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 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 Numerical Schemes Property
Discretisation in time 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 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 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 Numerical Schemes Property Discretisation TimeSpace FD FE Time continuous Euler scheme LW LW-FE LW-TG
Discretisation in time Numerical Schemes Property
Discretisation in time Propagation of a cosine profile LW LW-FE LW-TG 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:
Extension to non linear convection Discretisation in time 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
Extension to non linear convection Discretisation in time 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 Remarks: In the case of a scalar equation (and only) the third order time derivative may be written in the following compact form: The consistent mass matrix depends of the unkown
Extension to multi dimensional problems Discretisation in time Let consider the following hyperbolic equation The time derivatives may be expr