Schedule for: 18w5148 - Adaptive Numerical Methods for Partial Differential Equations with Applications

Arriving in Banff, Alberta on Sunday, May 27 and departing Friday June 1, 2018
Sunday, May 27
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (Corbett Hall Lounge (CH 2110))
Monday, May 28
07:00 - 08:45 Breakfast
Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
08:45 - 09:00 Introduction and Welcome by BIRS Staff
A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.
(TCPL 201)
09:00 - 10:00 Mikhail Shashkov: Adaptive reconnection-based Arbitrary Lagrangian-Eulerian method
We present a new adaptive reconnection-based Arbitrary Lagrangian Eulerian method - A-ReALE. The main elements of an A-ReALE method are: An explicit Lagrangian phase on arbitrary polygonal mesh in which the solution and positions of grid nodes are updated; a rezoning phase in which a new grid is defined - both number of cells and their locations as well as connectivity (based on using Voronoi tessellation) of the mesh are allowed to change; and a remapping phase in which the Lagrangian solution is transferred onto the new grid. The design principles of A-ReALE method can be summarized as follows. First, it is using monitor (error indicator) function based on gradient or Hessian of some flow parameter(s), which is measure of interpolation error. Second, using equidistribution principle for monitor function for creating of adaptive mesh. Third, using weighted centroidal Voronoi tessellation as a tool for creating adaptive mesh. Fourth, we modify raw monitor function - we scale it to avoid very small and very big cells and smooth it to create smooth mesh and allow to use theoretical results related to weighted centroidal Voronoi tessellation. We present all details required for implementation of new adaptive ReALE methods and demonstrate their performance in comparison with standard ReALE method on series of numerical examples. This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The authors gratefully acknowledge the partial support of the US Department of Energy Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research and the partial support of the US Department of Energy National Nuclear Security Administration Advanced Simulation and Computing (ASC) Program. This is joint work with W. Bo.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Andrew McRae: Mesh adaptivity on the sphere using optimal transport, and a moving mesh scheme for the nonlinear shallow water equations
In the first part of this talk, I discuss the generation of meshes adapted to a prescribed scalar 'monitor' function. This is done through equidistribution, so that the volume of a cell is inversely proportional to the monitor function. We supplement this with an optimal transport condition, which aids with mesh regularity, and guarantees existence and uniqueness of such a mesh. The resulting mesh can be obtained by solving a Monge-Ampère equation, a scalar nonlinear elliptic PDE. This optimal transport also approach generalizes naturally from Euclidean space to manifolds such as the sphere. In the second part of this talk, I discuss the integration of moving mesh adaptivity into a finite element shallow water model, in the wider context of the need for global numerical weather prediction models that can resolve small-scale dynamic features. We do this by modifying the governing fluid equations so they are solved in a frame relative to the moving mesh. The finite element discretization is based on a 'compatible', or 'mimetic', approach, in which the finite element spaces are linked by differential operators. The degrees of freedom correspond not just to point values, but also to fluxes and densities, which complicates the modifications that are required. This is joint work with Chris Budd (Bath) and Jemma Shipton and Colin Cotter (Imperial).
(TCPL 201)
11:30 - 13:00 Lunch
Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
13:00 - 14:00 Guided Tour of The Banff Centre
Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.
(Corbett Hall Lounge (CH 2110))
14:00 - 14:20 Group Photo
Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!
(TCPL Foyer)
14:20 - 15:00 JF Williams: Applications of moving mesh methods in rigorous computing
This talk will introduce the field of rigorous computing and explain some of the problems it solves and tools it requires in a classical fixed grid context. We will then consider continuation problems where moving meshes are particularly useful. r-adaptive methods provide numerical efficiency and also simplify some of the topological technicalities in this approach.
(TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:10 Matthew Hubbard: Continuous and discontinuous approaches to moving mesh finite elements
The moving mesh finite element method of Baines, Hubbard and Jimack [1] determines mesh velocities by building the underlying PDE into a monitor conservation principle and recovering an approximation in a manner which preserves the original monitor distribution (so an equidistributed mesh remains so as time progresses). The first part of this talk will outline this method and show some examples which illustrate its ability to track interfaces accurately for implicit moving boundary problems [2]. In theory, it is possible to drive the mesh movement in this algorithm using other monitors, e.g. arc-length, which could be used to distribute mesh nodes more effectively in the interior of the domain. Initial attempts to implement this as a continuous Galerkin finite element method have lacked robustness, so the second part of the talk will outline a discontinuous Galerkin version of the algorithm, capable of stabilising the approximation of the hyperbolic terms introduced when the mesh nodes no longer move according to a physically meaningful conservation principle. Initial results will be shown to demonstrate the feasibility of the approach and the extension to higher orders of accuracy will be discussed. I would particularly like to acknowledge the contributions to this work by Prof Mike Baines (University of Reading), Prof Peter Jimack (University of Leeds) and Mr Tom Radley (University of Nottingham). References: [1] M.J.Baines, M.E.Hubbard and P.K.Jimack, A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl. Numer. Math. 54:450--469, 2005. [2] M.J.Baines, M.E.Hubbard and P.K.Jimack, Velocity-based moving mesh methods for nonlinear partial differential equations, Commun. Comput. Phys. 10(3):509--576, 2011.
(TCPL 201)
16:10 - 16:50 Agnieszka Miedlar: Reducing FEM eigenvalue/eigenvector computations via p-hierarchical enrichment
Although adaptive approximation methods have gained a recognition and are well-established, they frequently do not meet the needs of real world applications. In this talk we present a hierarchically enhanced adaptive finite element method for PDE eigenvalue problems. Starting from the results of Grubišić and Ovall on the reliable and efficient asymptotically exact a posteriori hierarchical error estimators in the self-adjoint case, we explore the possibility to use the enhanced Ritz values and vectors to restart the iterative algebraic procedures within the adaptive algorithm. Using higher order hierarchical polynomial finite element bases, as indicated by Bank and by Ovall and Le Borne, our method generates discretization matrices whose compressions onto the complement of piecewise linear finite element subspace (in the higher order finite element space) are almost diagonal. This construction can be repeated for the complements of higher (even) order polynomials and yields a structure which is particularly suitable for designing computational algorithms with low complexity. We present some preliminary numerical results for both the symmetric as well as the nonsymmetric eigenvalue problems. This is a joint work with Luka Grubišić and Jeffrey S. Ovall.
(TCPL 201)
16:50 - 17:30 Suzanne Shontz: A parallel variational mesh quality improvement method for distributed memory machines
There are numerous scientific applications which require large and complex meshes. Given the explosive growth in parallel computer architectures, ranging from supercomputers to hybrid CPU/GPU architectures, there has been a corresponding increase in interest in parallel computer simulations. For computational simulations involving the above applications, algorithms, which generate the mesh and manipulate it in parallel, are required. In particular, parallel mesh quality improvement is required whenever meshes of low quality arise in such simulations. In this talk, we describe our parallel variational mesh quality improvement method designed for distributed memory machines. The method is based on the sequential variational mesh quality improvement method of Huang and Kamenski. Although most mesh quality improvement methods directly minimize an objective function that explicitly specifies the mesh quality, Huang and Kamenski use the Moving Mesh PDE (MMPDE) method to discretize and find the minimum/maximum of a meshing functional, which is related to the mesh quality in an implicit manner. We solve the resulting ODE in parallel, which yields a mesh with a better quality. To solve the ODE in parallel, the mesh is first partitioned into regions using METIS. Next, the ODE is solved in parallel on the interior nodes of each mesh sub region. For nodes belonging to partition boundaries, i.e., those that are shared among cores, the algorithm calculates a partial ODE solution. To complete the solution at the shared nodes, a reduction operation using non-blocking MPI collective communication is performed. In order to overlap communication with computation, we experiment with various strategies for data organization. We test the performance of our method for up to 128 cores on tetrahedral meshes with up to 160M elements. Excellent strong scaling results and typical weak scaling results for unstructured meshes are obtained. This talk represents joint work with Maurin Lopez Varilla (Husky Injection Molding) and Weizhang Huang (University of Kansas).
(TCPL 201)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Tuesday, May 29
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:00 John Mackenzie: An adaptive moving mesh method for geometric evolution laws and bulk-surface PDEs
In this talk I will consider the adaptive numerical solution of a geometric evolution law where the normal velocity of a curve in two-dimensions is proportional to its local curvature as well as a general non-geometric driving force. An interface tracking approach is used which requires the generation of a moving mesh. It is well known for this class of problems that moving mesh nodes purely in the normal direction can lead to numerical instabilities and hence some form of tangential mesh movement is necessary. We have developed an adaptive moving mesh approach to distribute the mesh points in the tangential direction using a moving mesh PDE. It will be shown that the resulting meshes evolve smoothly in time and are well adjusted to resolve areas of high curvature. Experiments will be presented to highlight the improvement in accuracy obtained using the new method in comparison with uniform arc-length mesh distributions. We will also discuss the use of the evolving adaptive curve mesh in the adaptive generation of bulk meshes for the solution of bulk-surface PDEs in time-dependent domains. The moving mesh approach will then be applied to a range of problems in computational biology including image segmentation, cell tracking and the modelling of cell migration and chemotaxis. This is joint work with Micheal Nolan, Christopher Rowlatt (Mathematics and Statistics Department, University of Strathclyde) and Robert Insall (Beatson Institute for Cancer Research, Glasgow).
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Yanqiu Wang: Finite element method on polygonal meshes
We start with a brief introduction to the conforming finite element method on polygonal/polyhedral meshes using the generalized barycentric coordinates. Then we present our recent work in this direction on the construction of a H(div)/H(curl) conforming element and a Crouzeix-Raviart type non-conforming element. Finally, we discuss how to develop an alignment and an equidistribution quality measures for polytopal meshes, and how to build a Moving Mesh PDE (MMPDE) moving mesh method based on these mesh quality measures. The work on H(div)/H(curl) conforming element is done in collaboration with Dr. Wenbin Chen. The work on mesh quality and moving mesh method is done in collaboration with Dr. Weizhang Huang.
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:00 Avary Kolasinski: A surface moving mesh method based on equidistribution and alignment​​
Given a mesh on a surface, our goal is to improve the quality of the mesh using a moving mesh method. To this goal, we will construct a surface moving mesh method based on mesh equidistribution and alignment conditions. We will then discuss several proven advantages of this surface moving mesh approach. Finally, we will study various numerical examples using both the Euclidean metric and a Riemannian​ metric.
(TCPL 201)
14:00 - 14:30 Shaohua Chen: Numerical simulations to solutions of damped p-system in the whole space
In this talk, we numerically solve the Cauchy problem for the hyperbolic p-system with time-gradually-degenerate damping term in the whole space. We map the whole space into a bounded domain and use both fixed non-uniform mesh and moving mesh methods to obtain a solution which converges to a steady state together with some theoretical analysis.
(TCPL 201)
14:30 - 15:00 Christina C. Christara: Adaptive and non-adaptive spline collocation methods for a discontinuous diffusion PDE with application to brain cancer growth
Error analysis and convergence results for PDE discretization methods are normally based on the assumption, among other, that the PDE coefficients are continuous functions. When PDE discretization methods are applied to PDEs (BVPs or IVPs) with discontinuous coefficients, numerical results indicate that the standard convergence orders are typically not observed, and, even more, convergence is not guaranteed. We consider spline collocation PDE discretization methods and their application to a discontinuous diffusion PDE, modelling brain cancer growth, with the discontinuity of the diffusion coefficient arising from the different properties of the white and grey matters of the brain. We consider techniques based on adaptive grids and the approximation of the discontinuous coefficients by continuous ones, and techniques based on adjusting the basis functions so that they satisfy appropriate discontinuity conditions. We present numerical results highlighting the strengths and weaknesses of different approaches. Joint work with Paul Muir.
(TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:10 Erik Van Vleck: Bistable traveling waves under discretization: moving meshes
In this talk we consider the impact of discretization on traveling waves of bistable reaction-diffusion equations. Both temporal discretization by Backward Differentiation Formulas (BDF) methods and spatial discretizations by moving mesh methods are analyzed. Special attention is paid to propagation failure in one space dimension and to discretization induced anisotropy in higher space dimensions.
(TCPL 201)
16:10 - 16:50 Kelsey DiPietro: Monge-Ampére methods for fourth order PDEs and applications to elastic interface problems
We present a robust moving mesh finite difference method simulation of fourth order nonlinear PDEs describing elastic-electrostatic interactions in two dimensions. We use and extend the Parabolic Monge-Amp ́ere methods developed by Budd and Williams [1] to solve a fourth order PDE with finite time singularity. A key feature in our implementation is the generation of a high order transformation between computational and physical meshes that can accommodate the high order derivatives in the PDE. The PDE derived from a plate contact problem develops finite time quenching singularities at discrete spatial location(s). The moving mesh method dynamically resolves these temporally forming singularities, while preserving the underlying length scales of the problem. We show that the PMA resolves the singularities to high accuracy and gives strong evidence of self similarity near blow up. We briefly discuss the prediction of the touchdown profile using the skeleton theory from [3]. We numerically predict the skeleton set for a variety of domains, including domains with cutouts and non-convex domains. These predictions and verification of the skeleton theory are made using variational moving mesh methods developed in [2]. Accurately resolving singularities on general domains motivates recent work in extending the parabolic Monge-Amp ́ere equation to problems with curved domains. We utilize the optimal transport methods in [4] paired with the PMA to efficiently solve PDEs on curved domains using finite difference methods on a fixed, uniform computational domain. We present preliminary results of this method for semi-linear blow-up problems, sharp interface and prescribed moving boundary problems in curved two dimensional regions. References [1] C.J. Budd and J.F. Williams. Moving mesh generation using the parabolic Monge-Ampere equation. SIAM Journal on Scientific Computing, 31(5):3438-3465, 2009. [2] W. Huang and L. Kamenski. A geometric discretization and a simple implementation for variational mesh generation and adapation. J. Compt. Phys., 301:322-337. 2015. [3] A.E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor. SIAM Journal On Applied Mathematics, 72(3):935-958, 2012. [4] J. Benamou, B. Froese, A. Oberman. Numerical solution of the optimal transportation problem using the Monge Ampere Equation. J. Compt. Phys., 260:107-126,2014.
(TCPL 201)
16:50 - 17:30 Natalia Kopteva: A posteriori error estimation on anisotropic meshes
Our main goal in this talk is to present residual-type a posteriori error estimates in the maximum norm, as well as in the energy norm, on anisotropic meshes, i.e. we allow mesh elements to have extremely high aspect ratios [2, 4]. The error constants in these estimates are independent of the diameters and the aspect ratios of mesh elements. Note also that, in contrast to some a posteriori error estimates on anisotropic meshes in the literature, our error constants do not involve so-called matching functions (that depend on the unknown error and, in general, may be as large as mesh aspect ratios). To deal with anisotropic elements, a number of technical issues have been addressed in [2, 4]. For example, an inspection of standard proofs for shape-regular meshes reveals that one obstacle in extending them to anisotropic meshes lies in the application of a scaled traced theorem when estimating the jump residual terms (this causes the mesh aspect ratios to appear in the estimator). For maximum norm estimates, the analysis also employs sharp bounds on the Green’s function from [1]. For the estimation in the energy norm, a special quasi-interpolation operator is constructed on anisotropic meshes, which may be of independent interest [4]. We shall also touch on that certain perceptions need to be adjusted for the case of anisotropic meshes. In particular, it is not always the case that the computed-solution error in the maximum norm is closely related to the corresponding interpolation error [3]. References: [1] A. Demlow and N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems, Numer. Math., 133 (2016), 707-742 [2] N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, SIAM J. Numer. Anal., 53 (2015), 2519-2544 [3] N. Kopteva, Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations, Math. Comp., 83 (2014), 2061-2070 [4] N. Kopteva, Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes Numer. Math., 137 (2017), 607-642. [5] N. Kopteva, Fully computable a posteriori error estimator using anisotropic flux equilibration on anisotropic meshes, (2017), arXiv:1704.04404.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Wednesday, May 30
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:00 Huazhong Tang: A moving mesh method for hyperbolic conservation laws
We develop an efficient moving mesh algorithm for 1D and 2D hyperbolic systems of conservation laws. It is formed by two independent parts: PDE evolution and mesh-redistribution. The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula. The iteration for the mesh redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step. In 1D, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several 1D and 2D test problems are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Jianxian Qiu: A moving mesh discontinuous Galerkin method for hyperbolic conservation laws
In this presentation, a moving mesh discontinuous Galerkin (DG) method is developed for the numerical solution of hyperbolic conservation laws. The method combines the DG method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of mesh partial differential equations. The mesh is a nonuniform mesh that is sparse in the regions where the solution is smooth and more concentrated near discontinuities. The method can not only achieve the high order in the smooth region, but also capture the shock well in the discontinuous region. For the same number of grid points, the numerical solution with the moving mesh method is much better than ones with the uniform mesh method. Numerical examples are presented to show the accuracy and shock-capturing of the method. This talk is based on a joint work with Dongmi Luo and Weizhang Huang.
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 17:30 Free Afternoon (Banff National Park)
17:30 - 19:30 Dinner (Vistas Dining Room)
Thursday, May 31
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:00 Jens Lang: Adaptive moving meshes in large eddy simulation for turbulent flows
In the last years considerable progress has been made in the development of Large Eddy Simulation (LES) for turbulent flows. The characteristic length scale of the turbulent fluctuation varies substantially over the computational domain and has to be resolved by an appropriate numerical grid. We propose to adjust the grid size in an LES by adaptive moving meshes. The monitor function, which is the main ingredient of a moving mesh method, is determined with respect to a quantity of interest (QoI). These QoIs can be physically motivated, like vorticity, turbulent kinetic energy or enstrophy, as well as mathematically motivated, like solution gradient or some adjoint-based error estimator. The main advantage of mesh moving methods is that during the integration process the mesh topology is preserved and no new degrees of freedom are added and therefore the data structures are preserved as well. We will present results for real-life engineering and meteorological applications.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Paul Zegeling: Optimal transformation-based adaptive grids
The talk consists of three parts. In the first part, I discuss stationary optimal grids for singularly perturbed boundary-value problems. In particular, a three-point finite difference nonuniform grid is derived of the optimal order six. It is an extension of the well-known results that reach fourth-order accuracy (`supra-convergence’) by choosing a special monitor function. How to apply this to nonlinear models? Next, we treat transformation-based grids for time-dependent PDEs. An optimal time-dependent transformation for monotone traveling wave solutions will be proposed. This type of transformations is related to a perturbed method of characteristics. Ideas are given to deal with non-monotone solutions as well. Can this be extended to two space dimensions? In the final part, fractional-order differential equations are introduced. These models play an increasingly important role in physics, biology and finance. How can we develop efficient adaptive grids for such cases? What are the main differences with integer-order DEs? This is joint work with my PhD students S. Iqbal and H. Zhou.
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:00 Hormoz Jahandari: Combining h- and r-adaptivity for finite-element models with jumping coefficients
Discontinuous (jumping) coefficients often appear in modelling problems where the computational domain represents inhomogeneous media. An example of this is the geophysical electromagnetic (EM) modelling problem, where these jumps occur at interfaces that separate regions with different conductivities. These interfaces, along with other problem features, such as singular EM sources and pointwise solution observations, motivate mesh refinement to achieve good accuracy. The goal of this study is to investigate the combined application of h- and r-refinement to reduce numerical error in the modelling of EM data. For simplicity, aspects of this hr-adaptivity are explored in 1D. The steady-state diffusion and Helmholtz equations (which are commonly solved for the EM scalar and vector potentials, respectively) constitute the physical PDEs (PPDEs) here, while the r-refinement is based on an equidistribution principle. The PPDEs and the mesh PDE are solved alternately in an iterative manner to reduce an error estimate to a desired level. At each iteration, the old mesh is h-refined, the error estimate and its corresponding monitor function are updated and the r-refinement is performed. Various finite-element (FE) a posteriori global and local error estimates were examined: while FE residual-based estimates were cheaper to compute, hierarchical error estimates were found to be better indicators of the true errors. The solutions of adjoint problems of the PPDEs were used to construct local error estimates. These estimates were successfully used for goal-oriented refinement of the FE models. The talk is based on a joint work with Scott MacLachlan and Ronald Haynes.
(TCPL 201)
14:00 - 14:30 Ronald Haynes: Parallel PDE based mesh generators
To reduce the overhead involved with the solution of the coupled system of physical and mesh PDEs, we propose a family of parallel solvers based on domain decomposition approaches. In this short talk I will review theoretical results obtained for the convergence of Schwarz methods for the steady state equidistribution principle at both the continuous and discrete levels and for a particular MMPDE solved by Rothe’s method.
(TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:10 Paul Muir: B-spline adaptive collocation/Runge-Kutta software with interpolation-based spatial error estimation for the error-controlled numerical solution of PDEs
An essential component of a high quality numerical software package is a framework that provides an error-controlled computation of the approximate solution. This means that the package returns a numerical solution such that an associated error estimate satisfies a user-prescribed tolerance. This type of computation has two important advantages: (i) the user can have reasonable confidence that the numerical solution has an error that is consistent with the requested tolerance, and (ii) the cost of the computation will be consistent with the requested accuracy. In this talk, we introduce a new software package, called BACOLRI, for the error-controlled numerical solution of 1D PDEs. This code employs a high order B-spline collocation algorithm to discretize the spatial domain and then uses a high order error control Runge-Kutta package to solve the resulting system of time-dependent Differential-Algebraic Equations. BACOLRI estimates the spatial error of the approximate solution on each time-step using an interpolation-based scheme and employs adaptive mesh refinement to provide control of the spatial error. We will briefly survey the family of solvers from which BACOLRI has evolved, describe the algorithms implemented in BACOLRI, and review numerical results that demonstrate the superior performance of BACOLRI compared to other comparable solvers.
(TCPL 201)
16:10 - 16:50 Ray Spiteri: eBACOLI: a time- and space-adaptive multi-scale PDE solver
Multi-scale models are commonly used to mathematically describe the evolution of many interesting and important systems. They attempt to capture dynamics on multiple scales and integrate them into a common framework. One way to do this is via partial differential equations (PDEs) that do not depend on spatial derivatives to represent dynamics on small scales. Upon spatial discretization, these PDEs reduce to sets of ordinary differential equations (ODEs) at each discrete spatial point. eBACOLI is a numerical software package for solving multi-scale models consisting of parabolic PDEs coupled with PDEs that do not depend on spatial derivatives in one spatial dimension. eBACOLI features adaptive error control in the temporal and spatial domains. It uses a B-spline collocation method for the spatial discretization to yield a set of ODEs, which together with the boundary conditions, form a system of differential-algebraic equations (DAEs). These DAEs are then solved using DASSL. We demonstrate this applicability and efficiency of this software package on a number of examples, including the monodomain model of cardiac electrophysiology.
(TCPL 201)
16:50 - 17:30 Qin Sheng: Note on improved exponentially fitted adaptations for the numerical solution of singular reaction-diffusion equations
Many finite difference methods that involve spatial adaptation employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This constructive strategy has been proven to be extremely effective and easy-to-use in multiphysical computations. However, selections of core monitoring functions are often challenging and crucial to the computational success. This note concerns several different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first a few monitoring designs to be discussed are within the so-called direct regime, the rest belong to a newer category of the indirect type, which requires the priori knowledge of certain important solution features or characteristics. Some simulated examples will be presented to illustrate our study and conclusions. This note is based on recent collaborative work with M. A. Beauregard and J. L. Padgett.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
19:30 - 21:00 General discussion (TCPL 201)
Friday, June 1
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:30 Weiwei Sun: New Analysis on Galerkin FEMs for Nonlinear Parabolic PDEs
Linearized (semi)-implicit schemes are the most commonly-used approximations in numerical solution of nonlinear parabolic equations since at each time step, the schemes only require the solution of a linear system. However the time step restriction condition of schemes is always a key issue in analysis and computation. For many nonlinear parabolic systems, error analysis of Galerkin type finite element methods with linearized semi-implicit schemes in the time direction is established usually under certain time step condition $\tau \le h^{\alpha}$ for some $\alpha>0$. Such a time-step condition may result in the use of a very small time step and extremely time-consuming in practical computations. The problem becomes more serious when a non-uniform mesh or adaptive meshing is used. In this talk, we introduce a new approach to unconditional error analysis of linearized semi-implicit Galerkin FEMs for a large class of nonlinear parabolic PDEs.
(TCPL 201)
09:30 - 10:00 Hong Zhang: A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media
An adaptive moving mesh finite difference method is presented to solve a modified Buckley Leverett equation with a dynamic capillary pressure term from porous media. The effects of the dynamic capillary coefficient, the infiltrating flux rate and the initial and boundary values are systematically studied using a traveling wave ansatz and efficient numerical methods. Special attention is paid to the non-monotonic profiles. The governing equation is discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive time dependent monitor function with directional control is applied to redistribute the mesh grid in every time step, and a diffusive mechanism is used to smooth the monitor function. The behavior of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction is investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, a good mesh quality and a high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the numerical method.
(TCPL 201)
10:00 - 10:30 Lennard Kamenski: Mesh smoothing based on the MMPDE moving mesh method
We present a mesh smoothing algorithm based on the MMPDE moving mesh method based on the direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities are given in a simple, analytical matrix form. We further combine the moving mesh smoothing with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity, and utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical comparison with some publicly available mesh improving software is provided. This work is a collaboration with Weizhang Huang (University of Kansas), Hang Si (Weierstrass Institute), Franco Dassi (Università degli Studi di Milano-Bicocca), and Patricio Farrell (Weierstrass Institute).
(TCPL 201)
10:30 - 11:00 Coffee Break (TCPL Foyer)
11:30 - 12:00 Checkout by Noon
5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.
(Front Desk - Professional Development Centre)
12:00 - 13:30 Lunch from 11:30 to 13:30 (Vistas Dining Room)