We further attempt to extend these r-adaptive methods to degenerate field theories, for example the nonlinear Schrödinger equation. Therefore, our approach provides a new perspective on the geometric integration of Poisson systems, which often arise as semi-discretizations of some integrable nonlinear partial differential equations, for example the Toda or Volterra lattice equations, and play an important role in modeling of many physical phenomena (see.
Hamiltonian mechanics
We have focused on the most important aspects that are key to understanding the later chapters of this assignment. We refer the reader to [23] and [38] for proofs of these theorems and more information on Hamiltonian systems.
Symplectic integrators for Hamiltonian systems
Basic definitions
The numerical scheme for the Hamiltonian system (2.1.6) defined by the mapping Fh is of convergent order r if there exists an open set U ⊂T∗Q and constants C>0,. A numerical scheme for the Hamiltonian system (2.1.6) defined by the mapping Fh is called symplectic if Fh in canonical coordinates satisfies the condition.
Runge-Kutta methods
The idea of partitioned Runge-Kutta methods is to take two different Runge-Kutta methods and apply the first to the q variables, and the other to the p variables. The symplectic Euler method (2.2.5) is an example of a partitioned Runge-Kutta method, combining the implicit Euler schemeb1 =1,a11=1 with the explicit Euler scheme.
Backward error analysis
Lagrangian mechanics
If the Lagrangian L is regular, then the flow FtL ∶ T Q Ð→ T Q for the Lagrangian vector field Z is symplectic on T Q, that is (FtL)∗ΩL=ΩL for all t∈R. If the Lagrangian is hyperregular, then the first equation can be solved for ˙q in terms of q and p.
Variational integrators for Lagrangian systems
Discrete Mechanics
2.4.5) Using one of the transformations one can define the discrete Lagrange biform onQ×Q with ωLd= (F±Ld)∗Ω, which yields˜ in coordinates. Using the Legendre transforms we can move to the cotangent bundle and define the discrete Hamiltonian map ˜FLd∶T∗QÐ→T∗Qby.
Correspondence between discrete and continuous systems
Using (2.4.4), the discrete Euler-Lagrange equations (2.4.2) can be equivalently rewritten in the position-momentum formulation on T∗Q axis. The accuracy of this approximation can be measured by defining the order of accuracy of the discrete Lagrangian.
Variational partitioned Runge-Kutta methods
As discussed in Section 2.4.1, the discrete Lagrangian defines the discrete Hamiltonian map ˜FLinto the cotangent bundleT∗Q. If the Lagrangian L is regular, then it can be shown that the discrete Lagrangian Ld is of order r if and only if the associated discrete Hamiltonian map ˜FLd is of order (see [41]).
Constrained mechanical systems
Lagrangian and Hamiltonian descriptions of constrained systems
The corresponding extended action functional SC ∶ C(Q) × C(Rd) Ð→R is given by. 2.5.3) The relationship between the dynamics of a restricted system on N and an extended system on Q×Rd is given by the following theorem (see [38]). However, it is again useful to consider the dynamics of this confined system in embedded space.
Variational integrators for constrained systems
However, the exact discrete Lagrangian for a bounded system is not simply the standard exact discrete Lagrangian (2.4.9) restricted to the constraint submanifoldN×N, as this would be the action along an unbounded trajectory. Higher-order variational integrators for constrained systems can be constructed as in Section 2.4.3: a variational Runge-Kutta method for a constrained Lagrangian system described by a regular Lagrangian L is equivalent to a symplectic partitioned Runge- Kutta method applied to the restricted Hamiltonian equations (2.5.6).
Field theory and multisymplectic geometry
In particular, for a second-order field theory one can prove the multisymlectic form(n+2) ΩLis defined in J3Y and a similar formula of the multisymlectic form. The multisymlectic form formula for first-order field theories is derived in [39] and generalized for second-order field theories in [34].
Multisymplectic variational integrators
By analogy with the continuous case, let P be the set of solutions of the discrete Euler-Lagrange equations (2.7.3). 2.7.5) The discrete form formula (2.7.4) is in direct analogy with the multisymplectic form formula (2.6.9) which applies in the continuous case.
Moving mesh partial differential equations
Equidistribution principle
An example discretization of this MMPDE on a uniform mesh in the computational domain can be The choice of grid density function ρ(X) is usually problem dependent and the subject of much research.
Mesh smoothing
As a way to introduce the mesh speed, we can consider the gradient flow equation, which has the form. However, this is not practical in calculations, since x(X, t) does not directly specify the location of the mesh points in the physical domain.
Coupling the mesh equations to the physical equations
- Quasi-Lagrange approach
- Rezoning approach
- Reparametrized Lagrangian
- Spatial Finite Element discretization
- DAE formulation and time integration
- Example
- Backward error analysis
The main advantage of the alternative solution procedure is the fact that the mesh and physical equations are not coupled and each can be solved more efficiently. The interpolation of the physical solution is a crucial step for the success of this approach and often needs to be done in a special way.
Lagrange multiplier approach to r-adaptation
Reparametrized Lagrangian
We therefore conclude that the control-theoretic strategy, while providing a perfectly legitimate numerical method, does not take full advantage of the underlying geometric structures. Let us point out that although we used a variational discretization of the governing physical PDE, the mesh equations were coupled in a manner typical of existing r-adaptive methods (see section 3.3 and [7], [28]).
Spatial Finite Element discretization
Invertibility of the Legendre Transform
The mass matrix M˜N(y, X) is non-singular almost everywhere (as a function of the yi's and Xi's) and singular iffγi−1=γi for some i. We use linear combinations of the first two rows of the mass matrix to zero out the elements of the B1 block below the diagonal.
Existence and uniqueness of solutions
Note that, unlike the continuous case, the time evolution of the grid points Xi is governed by the equations of motion, i.e. Mass matrix singularities (4.2.11) have some similarities with mass matrix singularities encountered in the moving finite element method.
Constraints and adaptation strategy
- Global constraint
- Local constraint
Note that when discretizing the finite elements on a grid coinciding with previously selected points, this system reduces to the approach presented in section 4.1: we minimize the discrete action only with respect to yi and complete the completed equations with the constraints (4.2 .40). In Section 3.2, we discussed how some of the adaptive constraints of interest can be derived from certain partial differential equations based on the uniform distribution principle, such as equation (3.2.7).
DAE formulation of the equations of motion
In particular we show that some of the integrators obtained in Chapter 4 can be interpreted as multisymplectic variational integrators (see Section 2.7). The discrete configuration bundle is Y = X ×R, and for ease of notation, let the fiber elements Yjibe be denoted by yij.
Analysis of the Lagrange multiplier approach
Note that the constraint is essentially lynonholonomic, since it depends on the derivatives of the fields. The restricted version (see Section 2.5.2 and [41]) of the Discrete Euler-Lagrange equations takes the form.
Generating consistent initial conditions
Proceeding in this way, we obtained Xi(d) and y(d)i as the numerical solution to (6.2.2) for the original value ofα. It is also useful to use (4.2.57) to calculate the initial values of the Lagrange multipliersλi which can be used as initial guesses in the first iteration of the Lobatto IIIA-IIIB algorithm.
Convergence
Note that the Lagrange multiplier strategy was able to accurately capture the motion of the soliton with only 17 mesh points (that is, N = 15). The trajectories of the mesh points for several simulations are depicted in Figure 6.3.6 and Figure 6.3.7.
Energy conservation
Since we did not use adaptive time-stepping and did not apply any mesh smoothing technique (see Section 3.2.2), the mesh quality deteriorated over time in all simulations, eventually leading to mesh intersection, i.e. , two grid points collapsing or crossing each other. Time integration was performed with 4th-order Gauss (top), 4th-order Lobatto IIIA-IIIB (middle), and 5th-order nonsymplectic Radau IIA (bottom).
Computational cost
However, one step of the Lagrange multiplier strategy takes 1.3447s, while one step of the theoretical-control strategy requires 1.418s when using the 4th-order Gaussian method and 2.2247s when the 4th-order Lobatto IIIA-IIIB method is used. For example, for N = 22 the theoretical control strategy gives a more accurate solution than the uniform mesh simulation for N = 90, but one step of the theoretical control strategy takes 0.2542 s (4th order Gaussian) and 0.3417 s (4th order Lobatto IIIA-IIIB), while the uniform mesh simulation requires 0.4975 s and 0.6333 s, respectively.
Geometric setup
Equations of motion
Mµν(q)q˙ν =∂µH(q) (7.1.8) forms a system of first-order ODEs, where we assume that the one-dimensional antisymmetric matrixMµν(q) =∂µαν(q) −∂ναµ (q ) is invertible for allq∈Q. This reflects the fact that the evolution of the considered degenerate system takes place on the primary constraint N = FL(T Q) ⊊ T∗Q.
Symplectic forms
Using the Legendre transform (7.1.3), we can define the Lagrangian biform ˜ΩLonT Q by ˜ΩL=FL∗Ω (see Section 2.3), which in canonical coordinates˜ (qµ,q˙µ) is given by . 7.1.19). If we use qµ as coordinates on N, then the local representation of Ω˜N will be given by (7.1.13).
Symplectic flows
Veselov discretization and Discrete Mechanics
- Discrete Mechanics and exact discrete Lagrangian
- Singular perturbation problem
- Exact discrete Legendre Transform
- Example
- Variational error analysis
We can also find an analytical expression for the exact discrete Lagrangian (7.2.1) associated with (7.2.19) as. This defines the discrete Lagrangian map FLd ∶ Q×Q Ð→ Q×Q and the associated discrete Hamiltonian map F˜Ld ∶ T∗Q Ð→ T∗Q, as explained in Section 2.4.1.
Variational partitioned Runge-Kutta methods
VPRK methods as PRK methods for the ‘Hamiltonian’ DAE
Condition (7.3.7) can be tedious to verify, especially when the Runge-Kutta method used has many stages. For a non-split Runge-Kutta method, we have A =A¯, and condition (7.3.7) is satisfied if it is invertible, and the measure matrix M(q) =DαT(q)-Dα(q), as defined in Section 7.1.1, is U-invertible and the inverse is bounded.
When applied to differential-algebraic equations, the order of convergence of the Runge-Kutta method can be reduced (see. The Runge-Kutta method with coefficients aij and bi, which is used for the DAE system (7.3.27), retains its classical order of convergence.
Nonlinear α µ (q)
- Runge-Kutta methods
- Partitioned Runge-Kutta methods
With the exception of the midpoint rule (s = 1), we see that the order of convergence of the Gaussian methods is reduced. Comments on the symplecticity of these schemes are the same as for the Gaussian methods in section 7.3.3.1.
Numerical experiments
Kepler’s problem
However, since the primary constraint N is not invariant under this flow, a result similar to Corollary 7.3.5 does not hold, i.e. the current is not symplectic to N. Similar calculations were performed for the Lobatto IIIA-IIIB and Radau IIA methods, also with h=0.1.
Point vortices
The non-variational Radau IIA scheme gives an exact solution but shows gradual energy dissipation. In the case of Gaussian methods, the Hamiltonian remained practically constant - visible minor non-uniform fluctuations are due to rounding errors.
Lotka-Volterra model
It would be interesting to construct Hamiltonian variational integrators for multisymplectic PDEs by generalizing the variational characterization of discrete Hamiltonian mechanics. Variational integrators for systems with non-holonomic constraints have mostly been developed in the Lagrange-d'Alembert setting.
![Figure 7.4.7: Hamiltonian for the 1-stage (top), 2-stage (second ), and 3-stage (third ) Gauss, and the 3-stage Radau IIA (bottom) methods applied to the system of two point vortices with the time step h = 0.1 over the time interval [ 0, 5 × 10 5 ] .](https://thumb-ap.123doks.com/thumbv2/123dok/10417507.0/166.918.167.677.256.846/figure-hamiltonian-second-gauss-methods-applied-vortices-interval.webp)
Verner’s method of order 6
The 3-stage Radau IIA method of order 5
Gauss methods of order 2, 4, and 6
Lobatto IIIA-IIIB pair of order 2
Lobatto IIIA-IIIB pair of order 4
Lobatto IIIA-IIIB pair of order 6
