Taylor-Galerkin Schemes 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 Outline

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

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

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

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

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

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

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

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 Time Space FD Time continuous FE FD Euler scheme FE LW LW-FE LW-TG Discretisation in time

Numerical Schemes Property Discretisation in time

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

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

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

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

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

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

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

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

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

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

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

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

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

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