Convergence Analysis of Strang Splitting for Vlasov-Type Equations

Einkemmer, Lukas; Ostermann, Alexander

doi:10.1137/130918599

Cited by 40 publications

(57 citation statements)

References 18 publications

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“…where the first term was bounded by Theorem 4.9 in [11]. The two remaining terms can be bounded by Lemmas 2.6 and 2.7 to give the desired estimate.…”

Section: Finally Since E Is Given By (23) It Follows Thatmentioning

confidence: 96%

“…The choice of f k+1/2 is such as to retain second order in the nonlinear case while still only advection problems have to be solved in the numerical approximation (for more details see, e.g., [11]). Note that since e τ 2 B(f k ) can be represented by a translation in the velocity direction only (which has no effect on the computation of the electric field) we can use here (2.4d) f k+1/2 = S (A) f k .…”

Section: Time Discretizationmentioning

confidence: 99%

“…Since such an approximation does not result in a function that is continuous across cell boundaries, the methods which are employed to show convergence for Hermite or spline interpolation are not applicable. In this paper, we will show, using the abstract (time) convergence result in [11], that this scheme converges with order O τ 2 + h q + h q /τ . Our main result is stated in Theorem 2.9 below.…”

Section: Introductionmentioning

confidence: 97%

“…In the seminal paper [17], the convergence properties of Strang splitting for evolution equations were analyzed with the help of the variation-of-constants formula. This approach was recently extended to Vlasov-type equations in [11]. In [4,5,24,7] semi-Lagrangian methods are combined with Strang splitting.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Einkemmer¹,

Ostermann

2014

SIAM J. Numer. Anal.

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A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov-Poisson equations is provided. It is shown that under suitable assumptions the error is of order O(τ 2 + h q + h q /τ ), where τ is the size of a time step, h is the cell size, and q the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed.

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“…where the first term was bounded by Theorem 4.9 in [11]. The two remaining terms can be bounded by Lemmas 2.6 and 2.7 to give the desired estimate.…”

Section: Finally Since E Is Given By (23) It Follows Thatmentioning

confidence: 96%

Section: Time Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Einkemmer¹,

Ostermann

2014

SIAM J. Numer. Anal.

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“…The results presented in this section can be compared with the one in [10] for the Strang splitting, where however only compactly supported solutions are considered.…”

Section: Introductionmentioning

confidence: 99%

High-order Hamiltonian splitting for the Vlasov–Poisson equations

et al. 2016

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Abstract. We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nyström type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.

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Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications

Shekarpaz

Azari

2019

Numerical Methods Partial

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In this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions of subproblems, the solution of the original problem is computed. Moreover, the framework of strongly continuous semigroup (or C0‐semigroup) is employed in error analysis of operator splitting method for the inverse problem. The convergence of the proposed method is also investigated and proved. Finally, some numerical examples in one, two, and three‐dimensional spaces are provided to confirm the efficiency and capability of our work compared with some other well‐known methods.

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Convergence Analysis of Strang Splitting for Vlasov-Type Equations

Cited by 40 publications

References 18 publications

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

High-order Hamiltonian splitting for the Vlasov–Poisson equations

Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications

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