Related papers: Invariant meshless discretization schemes
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our…
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by…
Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
This article reports on the efficiency of a co-located diffuse approximation method coupled with a projection algorithm for the solution of two and three-dimensional incompressible flow equations. Three typical examples show the accuracy of…
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…
Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast…
We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which…
The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the…
In this paper, we propose an approach for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we focus on the \emph{maximal admissible…
A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves…
We provide a comprehensive overview of meshfree collocation methods for numerically approximating differential operators on continuously labeled unstructured point clouds. Meshfree collocation methods do not require a computational grid or…
We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the…
The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize…