Time-dependent Partial Differential Equations Research Articles

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.

Dynamical low-rank approximation: applications and numerical experiments

Dynamical low-rank approximation is a differential-equation-based approach to efficiently compute low-rank approximations to time-dependent large data matrices or to solutions of large matrix differential equations. We illustrate its use in the following application areas: as an updating procedure in latent semantic indexing for information retrieval, in the compression of series of images, and in the solution of time-dependent partial differential equations, specifically on a blow-up problem of a reaction-diffusion equation in two and three spatial dimensions. In 3D and higher dimensions, space discretization yields a tensor differential equation whose solution is approximated by low-rank tensors, effectively solving a system of discretized partial differential equations in one spatial dimension.

Strong stability preserving hybrid methods

This paper is concerned with the strong stability preserving (SSP) time discretizations for semi-discrete systems, obtained from applying the method of lines to time-dependent partial differential equations. We focus on the construction of explicit hybrid methods with nonnegative coefficients, which are a class of multistep methods incorporating a function evaluation at an off-step point. A series of new SSP methods are found. Among them, the low order methods are more efficient than some well known SSP Runge–Kutta or linear multistep methods. In particular, we present some fifth to seventh order methods with nonnegative coefficients, which have healthy CFL coefficients. Finally, some numerical experiments on the Burgers equation are given.

High-order linearly implicit two-step peer – finite element methods for time-dependent PDEs

Linearly-implicit two-step peer methods are successfully applied in the numerical solution of ordinary differential and differential-algebraic equations. One of their strengths is that even high-order methods do not show order reduction in computations for stiff problems. With this property, peer methods commend themselves as time-stepping schemes in finite element calculations for time-dependent partial differential equations (PDEs). We have included a class of linearly-implicit two-step peer methods in the finite element software Kardos. There PDEs are solved following the Rothe method, i.e. first discretised in time, leading to linear elliptic problems in each stage of the peer method. We describe the construction of the methods and how they fit into the finite element framework. We also discuss the starting procedure of the two-step scheme and questions of local temporal error control. The implementation is tested for two-step peer methods of orders three to five on a selection of PDE test problems on fixed spatial grids. No order reduction is observed and the two-step methods are more efficient, at least competitive, in comparison with the linearly implicit one-step methods provided in Kardos.

An Optimal-Order Error Estimate for a Family of ELLAM-MFEM Approximations to Porous Medium Flow

Mathematical models used to describe porous medium flow lead to coupled systems of time-dependent nonlinear partial differential equations, which present serious mathematical and numerical difficulties. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid-orientation effect. The ELLAM-MFEM time-stepping procedure, in which an Eulerian-Lagrangian localized adjoint method (ELLAM) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this paper we prove an optimal-order error estimate for a family of ELLAM-MFEM approximations.

Parallelization in time through tensor-product space–time solvers

In this Note, we extend the fast tensor-product algorithm for the simulation of time-dependent partial differential equations. We use the natural tensorization of the space–time domain to propose, after discretization, a set of independent problems, each one with the complexity of a single steady problem. This allows for an efficient parallel implementation that is already interesting on small architectures, but that can also be combined with standard domain-decomposition-based algorithms providing a further direction of parallelism on large computer platforms. Preliminary numerical simulations are presented for a one-dimensional unsteady heat equation. To cite this article: Y. Maday, E.M. Rønquist, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $(k+1)$-th order of accuracy when using piecewise $k$-th degree polynomials, under the condition that $k+1$ is greater than or equal to the order of the equation.

Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation

This paper deals with the numerical analysis of time dependent parabolic partial differential equation. The equation has bistable nonlinearity and models electrical activity in a neuron. A qualitative analysis of the model is performed by means of a singular perturbation theory. A small parameter is introduced in the highest order derivative term. This small parameter is known as singular perturbation parameter. Boundary layers occur in the solution of singularly perturbed problems when the singular perturbation parameter tend to zero. These boundary layers are located in neighbourhoods of the boundary of the domain, where the solution has a very steep gradient. Most of the conventional methods fails to capture this effect. A numerical scheme is constructed to overcome this discrepancy in literature. A rigorous analysis is carried out to obtain a-priori estimates on the solution of the problem and its derivatives. It is then proven that the numerical method is unconditionally stable. Convergence and stability analysis is carried out. A set of numerical experiment is carried out and it is observed that the scheme faithfully mimics the dynamics of the model.

Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena

In this paper, we review and comment upon recently derived results for time dependent partial differential equation (PDE) models, which have been used to describe the various fluid-structure interactions which occur in nature. For these fluid-structure PDEs, this survey is particularly focused on the authors' results of (i) semigroup wellposedness, (ii) stability, and (iii) backward uniqueness.

On the accuracy of operator splitting for the monodomain model of electrophysiology

Nonlinear systems of time dependent partial differential equations of real life phenomena tend to be very complicated to solve numerically. One common approach to solve such problems is by applying operator splitting which introduces subproblems that are easier to handle. The solutions of the subsystems are usually glued together by either the Godunov method (first-order) or the Strang method (second-order). However, the accuracy of such an approach may be very hard to analyse because of the complexity of the equations involved. The purpose of the present paper is to introduce another line of reasoning concerning the accuracy of operator splitting. Let us assume that a fully coupled and implicit discretization of the complete system has been developed. Under appropriate conditions on the continuous problem, such discretizations provide reasonable and convergent approximations. As in the continuous case, operator splitting can be utilized to obtain tractable algebraic subsystems. The problem we address in the present paper is to obtain a bound on the difference between the fully coupled implicit discrete solutions, u h, and the solutions, u h, s, obtained by applying operator splitting to these discrete equations. Suppose we know that u h converges to the analytical solution u as the grid is properly refined and, applying the results from this paper, that u h, s converges toward u h under grid refinement. Then, by the triangle inequality, also the splitting approximation u h, s converges toward the analytical solution u and convergence is thus obtained for an approximation that, from a practical point of view, is easier to compute.

An Algorithm for Solving Reactive Advection-Dispersion Problems

The physical problem of advection, dispersion, and reaction of a suite of chemicals is formulated mathematically as a set of coupled time-dependent nonlinear partial differential equations. Partially based on the Gear scheme for time discretization, the algorithm developed in this paper decouples equations into separate linearized problems. Then, using finite element approximation, these resulting linear advection-dispersion equations are further transformed, via a re-assembling of variational formulation, into an elliptic Stokes-like problem, which can be solved by an existing Stokes solver previously employed when solving the flow problem to produce a velocity background. The methodology is implemented in the simulation of transport and transformations of an electron donor, an electron acceptor, and active biomass. This application provides insights into biofilm evolution and pore clogging, and demonstrates mesh self-adjustment in the process of biofilm growth.

A numerical solution for nonlinear PDEs

In this paper, we solve system of time dependent partial differential equations (PDEs) by using pseudospectral method. Firstly, theory of application of spectral collocation method on system of time dependent partial differential equations presented. This method yields a system of ordinary differential equations (ODEs). Secondly, we use forth-order Runge-Kutta formula for the numerical integration of the system of ODE. we consider some examples to illustrate the performance of the method described.

A Schwarz Waveform Moving Mesh Method

An $r$-refinement (moving mesh) method is considered for solving time dependent partial differential equations (PDEs). The resulting coupled system, consisting of the physical PDE and a moving mesh PDE, is solved by a Schwarz waveform relaxation method. In particular, the computational space-time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of PDEs on each subdomain. Dirichlet boundary conditions are used to pass solution information between neighboring regions. The efficacy of this approach is demonstrated for some model problems. For problems where the solutions evolve on disparate time scales in different regions of the spatial domain, this approach demonstrates the significant savings in computational time and effort which are possible.

Analysis of Direct Three-Dimensional Parabolic Panel Methods

Adherence boundary conditions for time dependent partial differential equations, via Chorin algorithm, can be reduced to a parabolic problem with Robin–Fourier boundary conditions in the three-dimensional context. In the spirit of panel methods, one establishes an integral formulation whose key point is the estimation of the potential density, introducing a kind of panel method for tangential kinematic boundary conditions. This paper discusses explicit estimations of this density in the general case of an arbitrarily shaped three-dimensional body, which leads to a fast numerical scheme. An error analysis is also provided, involving body smoothness, the Hölder exponent of the density, and whether the body presents torsion or not.

Spectral Analysis of Nonsymmetric Quasi-Toeplitz matrices with Applications to Preconditioned Multistep Formulas

The eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time-dependent partial differential equations is analyzed and discussed. The matrices generated by the underlying numerical integrators are small rank perturbations of block Toeplitz matrices; circulant-like preconditioners based on the former are considered. The eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of Krylov iterative accelerators. Due to several reasons (lack of symmetry, band structure, and coefficients depending on the size) the classical approach based on smooth generating functions gives very little insight here. Therefore, to characterize the eigenvalues, a difference equation approach exploiting the band Toeplitz and circulant patterns generalizing the well-known results of Trench is proposed.

Fast computation of a viscoelastic deformable Earth model for ice-sheet simulations

AbstractThe model used by Lingle and Clark (1985) to approximate the deformation of the Earth under a single ice stream is adapted to the purposes of continent-scale ice-sheet simulation. The model combines a layered elastic spherical Earth (Farrell, 1972) with a viscous half-space overlain by an elastic plate lithosphere (Cathles, 1975). For the half-space model we identify a new mathematical formulation, essentially a time-dependent partial differential equation, which generalizes and improves upon the standard elastic plate lithosphere with relaxing asthenosphere model widely used in ice-sheet simulation. The new formulation allows a significantly faster numerical strategy, a spectral collocation method based directly on the fast Fourier transform. We verify this method by comparing to an integral formula for a disk load. We also demonstrate that the magnitudes of numerical errors made in approximating coupled ice-flow/Earth-deformation systems are significantly smaller than pairwise differences between several Earth models. Our implementation of the Lingle and Clark (1985) model offers important features of spherical, layered, self-gravitating, viscoelastic Earth models without the computational expense.

A moving mesh method with variable mesh relaxation time

We propose a moving mesh adaptive approach for solving time-dependent partial differential equations. The motion of spatial grid points is governed by a moving mesh PDE (MMPDE) in which a mesh relaxation time τ is employed as a regularization parameter. Previously reported results on MMPDEs have invariably employed a constant value of the parameter τ. We extend this standard approach by incorporating a variable relaxation time that is calculated adaptively alongside the solution in order to regularize the mesh appropriately throughout a computation. We focus on singular, parabolic problems involving self-similar blow-up to demonstrate the advantages of using a variable relaxation time over a fixed one in terms of accuracy, stability and efficiency.

The effect of imposing a corner condition on the evolution of two-dimensional crystal morphologies by surface diffusion with anisotropic surface free energies

Continuity of the chemical potential is required from the physics of the evolution of crystal morphology by surface diffusion and in equilibrium crystal shapes (ECS). With an anisotropic surface free energy, multiple ECS with corners have been reported and instability developed in computing evolution of crystal morphology. For a cornered ECS that minimizes the total surface free energy, a corner condition is needed. Hence, in addition to continuity of the chemical potential, we add the corner condition in the evolution of crystal morphology and ensure the condition is satisfied in the numerical computation of the evolving surfaces. The evolution is governed by a nonlinear 4th order time dependent partial differential equation. By enforcing the corner condition, we obtain better stability in the evolution, which leads to ECS having lower total surface free energies than those obtained previously without satisfying the corner condition. Our study shows that the corner condition plays an important role for the cases of critically and severely anisotropic surface free energies that yield surface corners and facets.

Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods. Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.

Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data

The computational approximation of solutions to nonlinear partial differential equations (PDEs) such as the Navier–Stokes equations is often a formidable task. For this reason, there is significant interest in the development of very low-dimensional models that can be used to determine reasonably accurate approximations in simulation and control problems for PDEs. Many concrete tests have been reported on in the literature that show that several classes of reduced-order models (ROMs) are effective, i.e., an accurate approximate solution can be obtained using very low-dimensional ROMs. Most of these tests involved problems for which the solution depends on only a single parameter appearing in the boundary data. The extension of the techniques used in those tests to the case of multiple parameters is not a straightforward matter. Here, we present, test, and compare two methods for treating, within the context of a class of ROMs, inhomogeneous Dirichlet-type boundary conditions that contain multiple parameters. In this study, we focus on the proper orthogonal decomposition (POD) approach to ROM; however, the issues and results discussed here in the POD context apply equally well to other ROM approaches.