Journal• ISSN: 2804-7214
ESAIM
EDP Sciences
About: ESAIM is an academic journal published by EDP Sciences. The journal publishes majorly in the area(s): Computer science & Finite element method. It has an ISSN identifier of 2804-7214. Over the lifetime, 52 publications have been published receiving 87 citations. The journal is also known as: [ESAIM] & [European series in applied and industrial mathematics. Modélisation mathématique et analyse numérique].
Topics: Computer science, Finite element method, Discretization, Convergence (economics), Nonlinear system
Papers
TL;DR: In this paper , a semi-implicit finite element projection method for the magnetohydrodynamic (MHD) equations is proposed and analyzed, and the energy stability of the scheme is theoretically proved.
Abstract: In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank–Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type [Van Kan, SIAM J. Sci. Statist. Comput. 7 (1986) 870–891] in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete L ∞ (0, T ; L 2 ) norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.
38 citations
TL;DR: In this paper , the authors consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive-and conservative time integrator schemes based on the center manifold theory.
Abstract: Since almost twenty years, modified Patankar–Runge–Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Nevertheless, until now, an analytic investigation of the stability of MPRK schemes is still missing, since the usual approach by means of Dahlquist’s equation is not feasible. Therefore, we consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive and conservative time integrator schemes based on the center manifold theory. We use this approach to investigate the Lyapunov stability of the second order MPRK22(α) and MPRK22ncs(α) schemes. We prove that MPRK22(α) schemes are unconditionally stable and derive the stability regions of MPRK22ncs(α) schemes. Finally, numerical experiments are presented, which confirm the theoretical results.
22 citations
TL;DR: Novel methods are proposed that overcome an order barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions.
Abstract: Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.
20 citations
TL;DR: In this paper , a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method was proposed for brain tumor growth.
Abstract: Abstract. In this article, we consider the Cahn-Hilliard-Brinkman-Ohta-Kawasaki tumor growth system, which couples the Brinkman flow equations in the porous medium and the Cahn-Hilliard type equation with the nonlocal Ohta-Kawasaki term. We first construct a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method in the time discretization, and strictly prove its unconditional energy stability. The optimal error estimate for the tumor interstitial fluid pressure is further obtained. Numerical results are also carried out to demonstrate the effectiveness of the proposed numerical scheme and verify the theoretical results. Finally, we apply the scheme to simulate the evolution of brain tumors based on patient-specific magnetic resonance imaging, and the obtained computational results show that the proposed numerical model and scheme can provide realistic calculations and predictions, thus providing an in-depth understanding of the mechanism of brain tumor growth.
17 citations
TL;DR: The sparse grid reconstructions offer a significant improvement on the statistical error of PIC schemes as well as a reduction in the complexity of the problem providing the electric field.
Abstract: In this article, we propose and analyse Particle-In-Cell (PIC)
methods embedding sparse grid reconstruction as those introduced in [1, 2].
The sparse grid reconstructions offer a significant improvement on the sta-
tistical error of PIC schemes as well as a reduction in the complexity of the
problem providing the electric field. Main results on the convergence of the
electric field interpolant and conservation properties are provided in this pa-
per. Besides, tailored sparse grid reconstructions, in the frame of the offset
combination technique, are proposed to introduce PIC methods with improved
efficiency. The methods are assessed numerically and compared to existing PIC
schemes thanks to classical benchmarks with remarkable prospects for three
dimensional computations.
13 citations
Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 3 |
| 2025 | 66 |
| 2024 | 59 |
| 2023 | 49 |
| 2022 | 64 |