ArticlePDF Available

Stability of Moving Mesh Systems of Partial Differential Equations

Authors:
Shengtai Li at Los Alamos National Laboratory
Shengtai Li
Linda Petzold at University of California, Santa Barbara
Linda Petzold
Download full-text PDF
Read full-text
Download citation

Abstract and Figures

Moving mesh methods based on the equidistribution principle are studied from the viewpoint of stability of the moving mesh system of differential equations. For fine spatial grids, the moving mesh system inherits the stability of the original discretized PDE. Unfortunately, for some PDEs the moving mesh methods require so many spatial grid points that they no longer appear to be practical. Failures and successes of the moving mesh method applied to three reaction-diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh PDE. Dept. of Computer Science and Army High Performance Computing Research Center, University of Minnesota, MN 55455. The work of this author was partially supported by the Army High Performance Computing Research Center and by ARO contract number DAAL03-92-6-0247. y Dept. of Computer Science and Army High Performance Computing Research Center, University of Minnesota, MN 55455. The work of this author was partially supporte...
Figures - uploaded by Linda Petzold
Author content
All figure content in this area was uploaded by Linda Petzold
Content may be subject to copyright.
Content uploaded by Linda Petzold
Author content
All content in this area was uploaded by Linda Petzold on Dec 14, 2012
Content may be subject to copyright.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
−3 −2 −1 0 1 2 3
−1
−0.5
0
0.5
1
1.5
−3 −2 −1 0 1 2 3
−1
−0.5
0
0.5
1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
solid line:exact solution, mark: numerical solution
new monitor for Fisher problem (N=41)
t= 0.0, −*−
=1e−3, −o−
=2d−3, −+−
=3d−3, −x−
... Here we just mention a few only for the case of Fisher's equation, because later mainly the difficulties of the solution of the heat and diffusionconvection-reaction equations will be discussed. A moving mesh method was used by Li et al. [29] to solve Fisher's equation with a reaction term much stronger than the diffusion term. As they obtained rather inaccurate results, their conclusion was that their method is not really appropriate for this equation. ...
... This is because when the u n+1 i values at the end of the current time step are calculated, only the values u n i and u n i±1 at the beginning of this timestep are used at the first stage while u n i and u n+1 i±1 are used in the second stage. The two exceptions are formulas (29) and (32), which demand slightly larger memory because they break this rule. Interestingly, we will see that these two formulas do not yield very accurate algorithms, thus they will not be present in the final selection. ...
New stable, explicit, second order hopscotch methods for diffusion-type problems
Article
Full-text available
  • Feb 2023
  • MATH COMPUT SIMULAT
The aim of this paper is to systematically construct and test novel odd–even hopscotch-type numerical algorithms solving the diffusion or heat equation. Among the studied explicit two-stage methods, some of them are unconditionally stable and have second order convergence rate in time step size, which is proved analytically as well. We apply the best methods to the nonlinear Fisher’s equation to demonstrate that they work also for nonlinear equations. Then, in order to examine the competitiveness of the new algorithms, we test them for the heat equation against widely used numerical solvers in cases where the media are strongly inhomogeneous, and thus the coefficients strongly depend on space. The results suggest that the new methods are significantly more effective than the widely used explicit or implicit methods, especially for extremely large stiff systems.
View
... Using established moving mesh methods ( [22,24,6,19], e.g.), this article starts developing novel moving and merging small spatial patches in order to e ciently resolve both the shocks and the otherwise smooth macro-scale. This novel moving patch scheme should readily generalize to multi-D space analogous to established moving meshes. ...
... 3. Adaptively move patches to best resolve the macro-scale. To move each patch, we implement on the macro-scale the methodology of adaptive moving meshes ( [22,24,6,19], e.g.). That is, the patch-center locations X j (t) are to vary in time according to a standard moving mesh method ( Figure 1.1(b)). ...
View
... The organization of this article is as follows. In Section 3, we present the general form of the exact solution of Fisher's equation and in Section 4, we describe the numerical experiment [28]. ...
... The case of l = 0, a = 1 and b = 2, one obtains the Burgers-Huxley equation and the case of l = 0, a = 1, b = 1, we have Fisher's equation. The exact solution of Equation (11) is described in Li et al. [28] as a scaled Fisher's equation in the form ...
Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation
Article
Full-text available
  • Jul 2022
In this study, we obtain a numerical solution for Fisher's equation using a numerical experiment with three different cases. The three cases correspond to different coefficients for the reaction term. We use three numerical methods namely; Forward-Time Central Space (FTCS) scheme, a Nonstandard Finite Difference (NSFD) scheme, and the Explicit Exponential Finite Difference (EEFD) scheme. We first study the properties of the schemes such as positivity, boundedness, and stability and obtain convergence estimates. We then obtain values of L 1 and L ∞ errors in order to obtain an estimate of the optimal time step size at a given value of spatial step size. We determine if the optimal time step size is influenced by the choice of the numerical methods or the coefficient of reaction term used. Finally, we compute the rate of convergence in time using L 1 and L ∞ errors for all three methods for the three cases.
View
... Using established moving mesh methods (Li et al. 1998, MacKenzie & Mekw 2007, Budd et al. 2009, Huang & Russell 2010, this article starts developing novel moving and merging small spatial patches in order to efficiently resolve both the shocks and the otherwise smooth macroscale. This novel moving patch scheme should readily generalise to multi-D space analogous to established moving meshes. ...
... To move each patch, we implement on the macro-scale the methodology of adaptive moving meshes (Li et al. 1998, MacKenzie & Mekw 2007, Budd et al. 2009, Huang & Russell 2010. That is, the patch-centre locations X j (t) are to vary in time according to a standard moving mesh method (Figure 1(b)). ...
Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation
Preprint
Full-text available
  • Aug 2021
The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.
View
... A refined way to control the mesh movement by the residual of the original equation is given in moving finite element method (MFEM) in [1,19], which is again strongly related to equidistribution principle. Based on the equidistribution principle the stability of the moving mesh system is studied for reaction-diffusion problems in [18]. In [26], an efficient moving mesh method coupled with semi-implicit explicit solver for the physical PDE is given such that in practice the resulting scheme behaves as a two-step predictor-corrector methods. ...
Modified equation based mesh adaptation algorithm for evolutionary scalar partial differential equations
Article
Full-text available
  • Dec 2021
  • NUMER METH PART D E
It is well known that on uniform mesh classical higher order schemes for evolutionary problems yield an oscillatory approximation of the solution containing discontinuity or boundary layers. In this article, an entirely new approach for constructing locally adaptive mesh is given to compute nonoscillatory solution by representative “second” order schemes. This is done using modified equation analysis and a notion of data dependent stability of schemes to identify the solution regions for local mesh adaptation. The proposed algorithm is applied on scalar problems to compute the solution with discontinuity or boundary layer. Presented numerical results show underlying second order schemes approximate discontinuities and boundary layers without spurious oscillations.
View
... In order to obtain the stability condition on monitor function, we analyze equi-distribution principle in terms of linearize perturbation. Li et al. discussed the same approach for the stability analysis of reaction-diffusion problems in their work [17]. Huang, Ren and Russell discussed the stability condition for moving mesh PDE (MMPDE) by the same approach in their work [13]. ...
A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution
Article
  • Jan 2021
In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.
View
View
An Adaptive Conservative Moving Mesh Method
Chapter
  • Jan 2022
We present an adaptive moving mesh strategy endowed with a mass conservation property for the numerical solution of the two-dimensional linear transport equation. We achieve this through the solution of an auxiliary moving mesh problem driven by a gradient based monitor function. The method we propose is of staggered type, hence requires an appropriate data transfer operator over different meshes. We make use of an L⁡2{{\operatorname L} ^{2}} projection operator that requires the construction of a supermesh for evaluation. We summarise extensive numerical experiments aimed at testing the robustness of the method and its ability to approximate challenging numerical features.
View
Numerical Integration of Partial Differential Equations
Chapter
  • Sep 2021
A very large range of efficient techniques are exposed for solving partial differential equations. Various physical systems illustrate the different PDEs. The mathematical characterization of PDEs is explained. The method of characteristics is exposed by means of physical examples. The finite differences are very detailed with many different schemes and applications in heat and mass transfer and transport in 1D and 2D. Automatic finite differences and irregular grids are commented. Spectral methods, including Galerkin’s and collocations, radial basis functions, are explained with applications on ODEs and PDEs. A moving grid is detailed with application on a realistic chromatography. The finite volumes are detailed in 1D with application in heat and mass transfer. The finite elements are also explained with their foundations and algorithms, with many applications especially in heat transfer up to 3D. The boundary element method is treated in detail with application in heat transfer up to 2D. All these methods are explained mathematically and illustrated numerically with physical examples.
View
An Eulerian Vlasov-Fokker–Planck algorithm for spherical implosion simulations of inertial confinement fusion capsules
Article
  • Feb 2021
  • COMPUT PHYS COMMUN
We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov-Fokker–Planck (VFP) simulation of inertial confinement fusion (ICF) capsule implosions. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma’s evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: (1) an elastic moving wall boundary; (2) a time-dependent Maxwellian Dirichlet boundary; and (3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems –the first-ever reported using a Vlasov-Fokker–Planck model.
View
Moving mesh methods for partial differential equations
Article
Full-text available
  • Jan 1986
The numerical solution of nonlinear partial differential equations is often complicated by large local gradients in the solution that evolve in time. As these gradients pass over a mesh point, the solution at that mesh point changes rapidly. When this happens, small time steps should be taken at these mesh points to accurately integrate the solution while elsewhere larger time steps would suffice. Thus, a moving front can result in a multirate system of differential equations at the mesh points and force a global small time step. If the mesh points are moved so the solution at the mesh points changes slowly, then then the same larger time step may be appropriate everywhere. Several strategies are described on how to best move the meshpoints to define a reference frame where the solution is slowly varying.
View
Adaptive grid generation
Article
Full-text available
  • Oct 1987
A review of adaptive grid generation is presented with an emphasis on the basic concepts and the interrelationship between the various methods. The concepts are developed in a multifaceted progressive sense with enough detail so as to instill an operative spirit for the methods. The operational capabilities come from an explicit display of the necessary formulas for algorithmic construction. While virtually all adaptive procedures are aimed at problems with rapid solution variations, our main concern is the construction of methods that are not fundamentally restricted by the choice of problem or solution algorithm. Moreover, to maintain a simple treatment for the computational data and to have access to many of the best solution algorithms, we consider coordinate transformations. As a consequence, particular attention is given to grid point motion that occurs in response to the influence from the solution data, regardless of how that data was obtained. After some introductory discussion on the utilization of solution data, the topic of grid point motion is addressed first in one dimension and then in higher dimensions. The basic equidistribution process is first seen from a dozen different viewpoints in one dimension. This is further amplified with the practical notions of precise coefficient specification, the attraction to a given grid, and the action of evolutionary forces. With the definition of the metric, the direct extension into higher dimensions is developed with curve-by-curve methods. This is followed by finite volume methods and variational methods. With the various movement strategies established in a multidimensional context, the next consideration is the temporal coupling of the movement with the solution algorithm. This is undertaken in the discussion of temporal aspects.
View
View
View
Stability of Computational Methods for Constrained Dynamics Systems
Article
  • Jan 1993
Many methods have been proposed for numerically integrating the differential-algebraic systems arising from the Euler–Lagrange equations for constrained motion. These are based on various problem formulations and discretizations. We offer a critical evaluation of these methods from the standpoint of stability. Considering a linear model, we first give conditions under which the differential-algebraic problem is well conditioned. This involves the concept of an essential underlying ODE. We review a variety of reformulations which have been proposed in the literature and show that most of them preserve the stability of the original problem. Then we consider stiff and nonstiff discretizations of such reformulated models. In some cases, the same implicit discretization may behave in a very different way when applied to different problem formulations, acting as a stiff integrator on some formulations and as a nonstiff integrator on others. We present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense.
View
The Regularization of Linear Differential-Algebraic Equations
Article
  • Jan 1996
Differential-algebraic equations naturally arise in numerous important applied contexts, e.g., electrical circuits, constrained dynamical motion and fluid dynamics. The authors show how the structures of the solution space for some typical problems can be illuminated through the introduction of various regularizations which often can be given natural physical interpretations.
View
Regularization of Nonlinear Differential-Algebraic Equations
Article
  • Mar 1994
  • SIAM J MATH ANAL
This paper illustrates how initial value problems for nonlinear differential-algebraic equations can be regularized, i.e., converted to tractable singularly perturbed problems, by appropriate introduction of a small positive parameter ε. The corresponding outer limit provides the desired solution, including consistent initial conditions. Examples are given for problems with indices one, two, and three.
View
Equidistributing principles in moving finite element methods
Article
  • Dec 1983
  • J COMPUT APPL MATH
Recently Miller and his co-workers proposed a moving finite element method based on a least squares principle. This was followed by a similar method by the present authors using a Petrov—Galerkin approach. In this paper the two methods are compared. In particular, it is shown that both methods move their nodes according to an approximate equidistributing principle. This observation leads to a criterion for the placement of the nodes. It is also shown that the penalty function designed by Miller may also be used with the Petrov—Galerkin method. Finally, numerical examples are given, illustrating the performance of the two methods.
View
A Survey of Dynamically-Adaptive Grids in the Numerical Solution of Partial Differential Equations
Article
  • Jan 1985
  • APPL NUMER MATH
The construction of dynamically-adaptive curvilinear coordinate systems based on numerical grid generation and the use thereof in the numerical solution of partial differential equations is surveyed, and correlations are made among the various approaches. These adaptive grids are coupled with the physical solution being done on the grid so that the grid points continually move in the course of the solution in order to resolve developing gradients, or higher variations, in the solution. Particular attention is given to systems using elliptic grid generation based on variational principles. It is noted that dynamic grid adaption can remove the oscillations common when strong gradients occur on fixed grids, and that it appears that when the grid adapts to the solution most numerical solution algorithms work well. Particular applications in computational fluid dynamics and heat transfer are noted.
View
On the Numerical Solution of Initial/Boundary-Value Problems in One Space Dimension
Article
  • Aug 1982
The numerical solution of initial/boundary-value problems of the form A(u, x, t)u/sub t/ + B(u, x, t)u/sub x/ = c(u, x, t) is considered. Particular emphasis is placed on the solution of problems with large gradients, e.g., shocks and boundary layers. A mesh-selection technique is described that accurately places points in regions where the solution is rapidly changing. This is accomplished by a transformation of the original equations to a new coordinate system. Finite difference solutions for two sample problems are calculated.
View