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A Structure-Preserving Semi-Lagrangian based bi-fidelity method for the Vlasov--Poisson system with uncertainties
Xiaofeng Cai, Jiaojiao Guo, Liu Liu
TL;DR
Repeated high-resolution VP simulations make uncertainty quantification costly. The paper pairs a conservative SL high-fidelity solver with a pressureless EP low-fidelity surrogate, and reports accurate, robust uncertainty propagation across deterministic and uncertain tests.
Problem
Uncertainty quantification for VP is costly because each realization can require an expensive high-resolution kinetic computation, especially in high-dimensional random spaces.
Source: §1 · pp. 1–2Method
The method pairs a Strang-splitting CSLDG VP high-fidelity solver with a second-order MUSCL pressureless EP low-fidelity solver for macroscopic observables.
Source: §3.3 · p. 9, §6 · p. 22Results
Experiments on deterministic and uncertain problems validate the method's accuracy, robustness, and computational efficiency, including multispecies and higher-dimensional settings.
Source: §6 · p. 22Takeaways & Limitations
Conservative, positivity-preserving CSL snapshots support bi-fidelity reconstruction of complex dynamics and uncertainty propagation.
Source: §6 · p. 22Momentum conservation is not guaranteed; under strong Landau damping, momentum errors oscillate and increase.
Source: §5.1.1 · p. 15
Abstract
from arXiv · showhide
Uncertainty quantification for the Vlasov--Poisson (VP) systems entails repeated dynamic simulations, incurring immense computational costs. We develop a structure-preserving Semi-Lagrangian (SL) b-fidelity approximation of macroscopic observables. The MUSCL is used as a low-fidelity solver for pressureless Euler--Poisson system, allowing for a more cost-effective exploration of the stochastic parameter space. A conservative SL scheme combined with high-order discontinuous Galerkin is used as a high-fidelity solver for collisionless VP system, whose mass conservation, positivity preservation, and low numerical dissipation provide a more reliable high-fidelity snapshots for bi-fidelity reconstruction. We further derive a conditional error bound that separates projection, discretization, and model errors. Numerical experiments for both deterministic and uncertain VP problems, including multispecies and higher-dimensional settings, demonstrate that the proposed framework achieves accurate and robust uncertainty propagation.
1. Introduction.
Repeated kinetic simulations make uncertainty quantification costly, motivating a bi-fidelity approach that pairs conservative SL snapshots for VP with a pressureless EP surrogate. The framework combines structure-preserving high-fidelity simulations, lower-cost parameter exploration, and error analysis.
1. Introduction.
- Repeated high-resolution kinetic simulations make uncertainty quantification expensive, especially for high-dimensional phase spaces and random parameter spaces.Particle-based methods can require many particles for accurate macroscopic quantities, while Eulerian costs grow with phase-space dimension.Source: §1 · pp. 1–2
- The proposed high-fidelity solver uses Strang-splitting CSL with high-order DG reconstruction to produce conservative, positivity-preserving snapshots with low numerical dissipation.These properties are presented as improving the reliability of snapshots for bi-fidelity reconstruction.Source: §1 · p. 2
- A pressureless EP model serves as the low-fidelity surrogate, retaining essential macroscopic behavior while eliminating velocity-space discretization.The framework uses the low-fidelity model to explore parameter space and selects representative samples for high-fidelity evaluation.Source: §1 · pp. 2–3
- The work establishes error analysis for the bi-fidelity approximation and demonstrates applicability to multispecies and higher-dimensional kinetic problems.The introduction identifies these settings as including complex phase-space structures.Source: §1 · p. 3
2. Microscopic and Macroscopic Models.
The section presents VP as the kinetic model and pressureless EP as its macroscopic counterpart. A cold-electron closure yields a low-cost model that retains leading moments while avoiding velocity-space discretization.
2.1. Vlasov–Poisson System.
- The VP system evolves a phase-space distribution under a self-consistent electric field, with random parameters representing uncertainty in the model inputs.The spatial domain is periodic, and the velocity domain is chosen large enough for the distribution to decay at its boundary.Source: §2.1 · pp. 3–4
2.2. The macroscopic fluid equations.
- Taking velocity moments of the kinetic equation gives mass, momentum, and energy balances, but the resulting macroscopic equations are unclosed because they involve higher velocity moments.The momentum equation contains a pressure tensor, while the energy flux involves still higher-order moments.Source: §2.2 · p. 4
- Under the cold-electron monokinetic closure, the pressureless EP system determines energy from mass and momentum and retains leading moments while eliminating velocity-space discretization.The model uses the same Poisson equation for the self-consistent electric field.Source: §2.2 · pp. 4–5
3. A structure-preserving SL bi-fidelity method.
The method pairs conservative CSLDG snapshots of the VP system with a lower-cost pressureless Euler–Poisson surrogate, then uses the surrogate to guide high-fidelity sampling and reconstruction. Its solvers use Strang splitting, MUSCL reconstruction, positivity preservation, and mass-conservative updates.
3.1. High-fidelity solver.
- The VP solver combines high-order spatial discretization with conservative semi-Lagrangian updates, positivity preservation, and strict mass conservation.The CSLDG method removes the CFL restriction while retaining these properties.Source: §3.1 · p. 5
3.1.1. CSLDG method.
- Strang splitting advances spatial advection and electric-field-driven velocity advection in consecutive substeps.The spatial sweep transports the distribution at velocity v; the velocity sweep uses the electric field as its advection velocity.Source: §3.1 · p. 5
- The CSLDG update traces upstream intervals, including when they cross multiple grid cells, and its zeroth-moment update recovers mass conservation.The method partitions the upstream interval along the background grid to calculate the update.Source: §3.1.1 · pp. 6–7
3.1.2. Positivity-preserving limiter.
- The positivity-preserving limiter maintains nonnegative cell averages when the prior solution is nonnegative, without compromising local mass conservation.The scheme's split operators also preserve total mass under the stated boundary and zero-mean-field conditions.Source: §3.1.2 · p. 8
3.2. Low-fidelity solver.
- The low-fidelity pressureless EP solver uses a finite-volume discretization with MUSCL reconstruction of density and velocity in primitive variables.The passage specifies forward Euler time discretization and a local Lax–Friedrichs flux.Source: §3.2 · p. 9
3. A structure-preserving SL bi-fidelity method.
- CSLDG high-fidelity snapshots and second-order MUSCL low-fidelity solutions provide the two model levels for the structure-preserving bi-fidelity method.The VP solver computes macroscopic moments from the kinetic model; the pressureless EP solver supplies its low-fidelity counterpart.Source: §3 · p. 5, §3.3 · p. 9
3.3. Bi-fidelity algorithm.
- The bi-fidelity algorithm explores parameter samples with the low-fidelity model, runs the high-fidelity model at selected samples, and combines those snapshots for new queries.Its effectiveness assumes an accurate low-dimensional approximation and sufficiently correlated low- and high-fidelity parametric structures.Source: §3.3 · pp. 9–10
4. Theoretical analysis.
The analysis bounds the difference between the high-fidelity solution and its bi-fidelity approximation by separating model discrepancy, projection error, and coefficient propagation error. The estimate is conditional on regularity and parameter assumptions and on the stated solver error bounds.
4.1. Notations and prerequisites.
- The prerequisites require uniformly bounded initial-data regularity, nonnegative data compactly supported in velocity, and an affine representer with coefficients in ℓβ for 0 < β < 1.The random parameter components are assumed to have compact support.Source: §4.1 · p. 11, §4.2 · p. 12
4.2. Main Error Estimate.
- Theorem 4.4 bounds the high-fidelity/bi-fidelity error by contributions from high-/low-fidelity discrepancy, low-fidelity projection, and coefficient propagation.The theorem states the estimate after K high-fidelity simulation runs, with constants independent of K, τ, and h.Source: §4 · p. 11, §4.2 · p. 12
- The high-/low-fidelity discrepancy includes high-fidelity discretization error, kinetic–macroscopic model error, and low-fidelity discretization error.The analysis draws on bounds for the VP-to-EP macroscopic deviation and for the CSLDG solver under the stated assumptions.Source: §4.1 · p. 11, §4.2 · p. 12
- The projection-error analysis uses the Kolmogorov K-width of the low-fidelity solution manifold and a truncated Legendre expansion.The stated projection bound relies on the affine representer condition for the random initial data.Source: §4.2 · pp. 12–13
5. Numerical experiments.
The experiments assess deterministic CSLDG accuracy and conservation, then test bi-fidelity approximation across single-component, two-component, and 2D1V uncertain VP problems. The reported comparisons show close agreement between bi-fidelity and high-fidelity results across these settings, while low-fidelity solutions can be inaccurate.
5. Numerical experiments.
- The deterministic tests examine spatial convergence, conservation, and fine phase-space structures; uncertain tests add random initial-condition parameters and assess bi-fidelity robustness and statistical convergence.The experiments also extend to electron-ion systems and higher-dimensional phase space.Source: §5 · p. 13
5.1.1. 1D1V.
- For P0-, P1-, and P2-CSLDG, spatial error decreases at approximately first-, second-, and third-order rates, matching theoretical order p + 1.The convergence study uses CFL = 0.01 and a time step chosen to make temporal error negligible.Source: §5.1.1 · p. 14, Table 1 · p. 14
- In Landau damping, P1-CSLDG resolves filamentary structures under strong damping and reproduces the reported electric-energy decay and growth rates.For weak damping, the decay rate matches γ = −0.153; for strong damping, energy decays at γ1 = −0.287 and later grows at γ2 = 0.085, in agreement with.Source: §5.1.1 · p. 15, Figure 1 · p. 15
5.1.2. 2D1V.
- For the deterministic multi-vortex two-stream instability, CSLDG resolves phase-space holes and intense filamentation, while mass and momentum errors remain close to machine precision.Electric-field energy grows exponentially before nonlinear saturation, then exhibits persistent multi-frequency oscillations.Source: §5.1.2 · pp. 16–17, Figure 3 · p. 17, Figure 4 · p. 17
5.2. Uncertain problem.
- The uncertain-problem tests use P1-CSLDG as the high-fidelity VP solver and second-order MUSCL as the low-fidelity pressureless Euler–Poisson solver.The reported setup uses 1000 random training samples and 100 test samples.Source: §5.2 · p. 18
5.2.2. Two-Component.
- With r = 30, the two-component bi-fidelity approximation reaches maximum accuracy of O(10^-4) across ρe, ue, ρi, and ui, while the low-fidelity baseline stagnates.At a representative spatial point, bi-fidelity solutions coincide with high-fidelity results; low-fidelity results deviate in magnitude and phase.Source: §5.2.2 · p. 20, Figure 7 · p. 21
5.2.3. 2D1V.
- In uncertain 2D1V, bi-fidelity reproduces high-fidelity spatial patterns and statistical characteristics for ρ and u, whereas low-fidelity misses essential structures.Mean and standard-deviation contours show visual agreement with high-fidelity results across the two-dimensional physical domain.Source: §5.2.3 · pp. 21–22, Figure 7 · p. 21, Figure 8 · p. 22
6. Conclusion.
The paper concludes that conservative semi-Lagrangian snapshots support bi-fidelity approximation of macroscopic observables for uncertain Vlasov–Poisson problems. Its error analysis and numerical experiments support the framework’s accuracy and robustness, while the conclusion identifies extensions for future work.
6. Conclusion.
- The framework combines a Strang-splitting conservative SL-DG high-fidelity solver with low-fidelity snapshots to approximate macroscopic observables and propagate uncertainty.The high-fidelity solver preserves conservation and positivity and resolves fine-scale phase-space structures.Source: §6 · p. 22
- The error analysis provides rigorous estimates that decompose the total error into several bounds analyzed individually.Source: §6 · p. 22
- The conclusion reports numerical validation of accuracy, robustness, and computational efficiency on deterministic and uncertain problems.Future work is directed toward non-splitting high-dimensional grids, multi-scale problems, and complex kinetic models.Source: §6 · p. 22
- Fig. 9 compares two-dimensional spatial distributions of low-, high-, and bi-fidelity solutions for a fixed sample.Source: Figure 9 · p. 23
- Fig. 10 displays the two-dimensional means and standard deviations of density and velocity for high-fidelity and bi-fidelity solutions.The caption assigns means to the upper panels and standard deviations to the lower panels, with r = 30.Source: Figure 10 · p. 23
SUPPLEMENTARY MATERIALS: A STRUCTURE-PRESERVING SEMI-LAGRANGIAN BASED BI-FIDELITY METHOD FOR THE VLASOV–POISSON SYSTEM WITH UNCERTAINTIES∗
The supplementary material details the modulated-energy argument and the Strang-splitting DG error decomposition. It states assumptions and bounds used to establish the supporting error estimates.
SM1. Proof of Lemma 4.1
- For smooth pressureless Euler–Poisson solutions and a nonnegative weak VP solution, the modulated-energy argument assumes finite time-integrated L∞ control of the reference flow’s symmetric strain.The reference density and velocity are assumed to lie in H^s, with s > d/2 + 1.Source: SUPPLEMENTARY MATERIALS: A… · p. 27
- The modulated total energy combines kinetic and electric-field errors, and its differential inequality is controlled by the reference flow’s strain before Grönwall’s inequality is applied.The supplement also states conservation of total VP mass.Source: SUPPLEMENTARY MATERIALS: A… · pp. 27, 29
SM2. Proof of Lemma 4.2
- The Strang-splitting DG error is decomposed into nonlinear-operator, propagated-error, projection, and single-step consistency components.The smooth-solution projection error is bounded by Ch^p+1; the prior-step projection component has the same stated bound.Source: SUPPLEMENTARY MATERIALS: A… · pp. 30–31
- The supplement combines the error components into a recurrence and invokes the discrete Grönwall lemma under a bounded initial projection error to obtain a global estimate.The local consistency component is tied to the cited local consistency theorem.Source: SUPPLEMENTARY MATERIALS: A… · pp. 31–32