اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDiscrete approximation of the minimizing movement scheme for evolution equations of Wasserstein gradient flow type with nonlinear mobility
203
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We propose a fully discrete variational scheme for nonlinear evolution equations with gradient flow structure on the space of finite Radon measures on an interval with respect to a generalized version of the Wasserstein distance with nonlinear mobility. Our scheme relies on a spatially discrete approximation of the semi-discrete (in time) minimizing movement scheme for gradient flows. Performing a finite-volume discretization of the continuity equation appearing in the definition of the distance, we obtain a finite-dimensional convex minimization problem usable as an iterative scheme. We prove that solutions to the spatially discrete minimization problem converge to solutions of the spatially continuous original minimizing movement scheme using the theory of $Gamma$-convergence, and hence obtain convergence to a weak solution of the evolution equation in the continuous-time limit if the minimizing movement scheme converges. We illustrate our result with numerical simulations for several second- and fourth-order equations.
قيم البحث
اقرأ أيضاً
We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient
flow of an entropy functional in the $L^2$-Wasserstein metric, the second is the Lagrangian nature, meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps. The resulting numerical scheme is entropy diminishing and mass conserving. Further, the scheme is weakly stable, which allows us to prove convergence under certain regularity assumptions. Finally, we present results from numerical experiments in space dimension $d=2$.
We recast the Calabi flow in DeGiorgis language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions,
and that sufficiently regular minimizing movements are smooth solutions to Calabi flow.
In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacl
e inside the computational domain is treated with a penalization method introducing a parameter $eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $epsilon$, which induces a small divergence and the time step $delta$t tend to zero with a proportionality constraint $epsilon$ = $lambda$$delta$t. Finally, when $eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{e}sum{e} Dans ce travail nous analysons un sch{e}ma de projection vectorielle (voir [1]) pour traiter le d{e}placement dun corps solide dans un fluide visqueux incompressible dans le cas o` u linteraction du fluide sur le solide est n{e}gligeable. La pr{e}sence de lobstacle dans le domaine solide est mod{e}lis{e}e par une m{e}thode de p{e}nalisation. Nous montrons la stabilit{e} du sch{e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $epsilon$ qui assure une faible divergence et le pas de temps $delta$t tendent vers 0 avec une contrainte de proportionalit{e} $epsilon$ = $lambda$$delta$t. Finalement nous montrons que leprob{`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{`i} ere immerg{e}e quand le param etre de p{e}nalisation $eta$ tend vers 0.
High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a numerical sch
eme with the order of total calculation cost $O(N log 2N)$ is proposed. As benchmark results, the relation between the numerical precision and the discretization unit size are demonstrated.
We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers
important cases such as Fokker-Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
