A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations
Summary (3 min read)
1. Introduction
- This paper constitutes the progression of a previous work [3] which focused on the steady solution of some classes of multidimensional partial differential equations (PDE) defined in spaces of dimension N (with N 1), and that is extended in the present work for treating multidimensional parabolic PDE.
- In [3] a new solution technique based on a tensor product representation was introduced.
- The authors would like to emphasize that they are not looking for a general numerical procedure for solving multidimensional PDE’s.
- Moreover, some recent works [9] on sparse grids prove its ability for integrating functions defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives towards the solution of multidimensional partial differential equations.
- Usual molecular descriptions make use of a configuration distribution function that is defined in a bounded domain and vanishes at its boundary.
2. Illustrating the solution strategy
- Fkj are not given in advance but they are computed adaptively using the algorithm described later.
- The construction of solution (Eq. (3)) consists of an iteration procedure involving, at each iteration n, three steps (see [3] for additional details).
- The functions Fk(n+1) are finally obtained by normalizing the functions R1, R2, . . . , RN+1 after convergence of the non-linear problem.
- Thus, in the steady case this ansatz leads to symmetric and positive definite linear systems (when the alternating direction strategy described in Section 3.2.1 applies) for the differential operators having these properties.
3.1. Solving the heat equation
- As seen previously, the construction of such solution involves, at each iteration n, a projection, convergence checking and enrichment steps that the authors now describe.
- Thus, Fj(x) (respectively,Gj(y) andHj(t)) and R(x) (respectively, S(y) and P(t)) are defined using a 1D finite element interpolation that, in their simulations, is assumed piecewise linear.
- The authors denote byN (respectively,M and L) the vector containing the value of the p (respectively, q and r) shape functionsNp(x) (respectively, Mq(y) and Lr(t)).
3.2.1. Efficient non-linear solvers
- In high-dimensional spaces, the discrete system (38) is strongly non-linear, and the convergence of non-linear solvers becomes a delicate task.
- In their previous work [3] a Newton strategy was successfully applied.
- That from their numerical experiments seems to be faster, simpler and more robust.
- These linear systems are solved within an iteration procedure until convergence.
- Other alternatives for solving this kind of strongly non-linear problems are being investigated, as is the case of the so-called asymptotic numerical technique [7].
3.2.2. Consistent time discretization
- Another important numerical issue is the discretization of the time derivative term in the variational formulation (18).
- If the authors consider the components of L as the usual piecewise linear shape functions, and perform the integral∫.
- This is equivalent (dividing by the time step ht) to the central finite difference which is not consistent with the physics (justifying the use of backward time derivatives).
- As the authors are discretizing the space-time variational formulation they cannot apply backward finite difference, and for this reason they propose to use an “upwind” Petrov–Galerkin formulation which leads to a backward time discretization.
- Thus, the time discretization is now fully consistent.
3.3. Numerical example
- The width of each representation of the 1D approximation functions illustrates the weight of that function, which is given by the corresponding α coefficient.
- The computed solution is in perfect agreement with the one obtained by using a classical finite element method.
3.4. Convergence analysis
- In order to conclude about the convergence rates of the proposed strategy the authors are considering a steady problem as well as a transient one whose large time solution coincides with the solution of the steady problem.
- (51) This problem is solved in different multidimensional spaces considering the separated representation and an increasing number of nodes for the functional approximation in each dimension (for the sake of simplicity the authors are considering that all the one-dimensional functional approximations are defined from the same number of nodes nn).
- The authors have reported in Fig. 3 the CPU time associated to each solution.
- The same slope of two is noticed proving that the CPU time increases in a similar way with the space dimension and with the number of degrees of freedom considered for the one-dimensional functional approximations.
- In these solutions only the number of nodes involved in the time discretization is increased.
4.1. Model equations
- Finally, the link between the statistical distribution of dumbbell configurations and the polymer stress τp is provided, for the 1D molecular configuration considered in this section, by τp = 〈h(q)q2〉 − 1 = ∫ Ψ (q)h(q)q2 dq− 1. (59) Remark 3. (i) Eq. (56) defines the time evolution of the molecule distribution function.
- The proposed method will be illustrated in homogeneous flows; the material derivative thus reduces to the partial derivative.
4.2. Problem discretization
- The use of the variable separation strategy requires null boundary and initial conditions.
- Due to both, the parabolic and convective character of Eq. (66), suitable stabilizations are required.
4.3. Extension to 1D multi-bead-spring models
- The technique described above can be applied for simulating the Multi-Bead-Spring models of polymer chains.
- The bead serves as an interaction point with the solvent and the spring contains the local stiffness information depending on local stretching (see [5] for more details).
- Eq. (75) can be solved using a separated representation.
5. Numerical results
- Fig. 6 shows the influence of the enrichment on the computed solution at tmax.
- The polymer stress can then be calculated as a function of time using Eq. (59).
- The authors must mention that the direct computation of the steady state is an issue for stochastic simulation techniques.
- The last example that the authors are considering in the present work concerns the transient 2D FENE model in a starting shear flows.
- (79) Fig. 14 depicts the three 2D space functions Fj(q) as well as the three associated time functions.
6. Conclusions and perspectives
- This paper proves the ability of the proposed strategy to compute transient solutions of multidimensional parabolic PDE’s with homogeneous boundary conditions admitting a separated representation.
- The novelty of the proposed technique justifies that many questions remain open: (i) the treatment of non-linear Fokker–Planck equations; (ii) optimal basis enrichment; (iii) analysis of complex flows involving non-homogeneous solution in the physical space; (iv) general initial conditions; (v) convergence analysis; (vi) stabilization of advection operators; . . ..
- The use of a wavelet approximation, could offer an optimal refinement via the multiresolution properties of wavelets (additional degrees of freedom are only required in regions where the wavelets coefficients are not small enough).
- The only remaining difficulty lies in the fact that nodes are assumed moving with the flow, which leads to highly distorted elements if one uses the finite element method (FEM) for solving momentum and mass balance equations.
- At present the stabilization of the advection operators has been performed by considering a SUPG technique.
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Citations
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Cites methods from "A new family of solvers for some cl..."
...An alternative, which allows to control the error global-in-time, is to apply iterative solvers to a space-time formulation [5,8,61,64,94]....
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Cites background from "A new family of solvers for some cl..."
...Key words: Model reduction, Evolution problems, Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), Generalized Spectral Decomposition, Separation of variables Preprint February 10, 2010 ha l-0 04 55 63 5, v er si on 1 - 10 F eb 2...
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Cites methods from "A new family of solvers for some cl..."
...This technique has already been used in the context of stochastic problems [19] and multidimensional problems [1, 2, 6]....
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...The PGD has been successfully applied to more general problems with different denominations: “Generalized Spectral Decomposition” for stochastic problems in [19], by replacing the time variable by a stochastic one ; “separated representation technique” for multivariable problems in [1, 2, 6], by adding new coordinates in the separated representation....
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References
578 citations
"A new family of solvers for some cl..." refers background in this paper
...Moreover, some recent works [9] on sparse grids prove its ability for integrating functions defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives towards the solution of multidimensional partial differential equations....
[...]
546 citations
"A new family of solvers for some cl..." refers background or methods or result in this paper
...13 and compared to the steady state simulation done in [3]....
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...Introduction This paper constitutes the progression of a previous work [3] which focused on the steady solution of some classes of multidimensional partial differential equations (PDE) defined in spaces of dimension N (with N 1), and that is extended in the present work for treating multidimensional parabolic PDE....
[...]
...In our previous work [3] a Newton strategy was successfully applied....
[...]
...The technique proposed in our former work [3] and extended in the present one to transient simulations, is, in our opinion, a suitable choice when dealing with highly multidimensional parabolic PDE’s with homogeneous boundary conditions....
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...These functions can be accurately represented using spherical coordinates or rectangular coordinates combined with a penalty technique as considered in [3]....
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393 citations
340 citations
"A new family of solvers for some cl..." refers background in this paper
...In this context, the sparse grids or sparse tensor product basis [6] are excellent candidates, even if it was argued in [1] its inability for treating highly multidimensional problems (N 20)....
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Q2. What are the future works in "A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. part ii: transient simulation using space-time separated representations" ?
For this purpose the authors consider that each node used in the discretization of the physical domain moves along its flow trajectory, allowing the integration of the advection operator of the Fokker–Planck equation using the method of characteristics. One possibility to circumvent this difficulty consists in using a meshless natural element method ( NEM ) [ 11 ] instead usual finite elements, because the accuracy of the NEM does not require any geometrical quality of the nodal distribution. Other fixed mesh techniques can be used, as the ones based on the characteristics-Galerkin scheme. In the case of general initial conditions, the authors can compute a separated representation of that initial condition ( as the sum of a product of terms involving the different coordinates ) using for example an alternating least squares technique [ 4 ].