Time-dependent Partial Differential Equations Research Articles

Adaptive energy preserving methods for partial differential equations

A framework for constructing integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The approach can be used with both finite difference and partition of unity methods, thereby including finite element methods. The schemes are then extended to accommodate r-, h- and p-adaptivity. To illustrate the ideas, the method is applied to the Korteweg–de Vries equation and the sine-Gordon equation. Results from numerical experiments are presented.

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.

Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media

The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations (ODEs), which include a single resonance linear Lorentz dispersion, and the nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. To design efficient, accurate, and stable computational methods, we apply high order discontinuous Galerkin discretizations in space to the hybrid PDE-ODE Maxwell system with several choices of numerical fluxes, and the resulting semi-discrete methods are shown to be energy stable. Under some restrictions on the strength of the nonlinearity, error estimates are also established. When we turn to fully discrete methods, the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms. To overcome this, novel strategies are proposed to treat the nonlinearity in our model within the framework of the second-order leap-frog and implicit trapezoidal time integrators. The performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves, and third-harmonic generation in soliton propagation.

Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations

This paper presents an arbitrary high-order accurate ADER Discontinuous Galerkin (DG) method on space–time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent partial differential equations for compressible dissipative flows : the compressible Navier–Stokes equations and the equations of viscous and resistive magnetohydrodynamics in two and three space-dimensions.The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell finite volume limiting procedure suitable for discontinuous Galerkin methods (Dumbser et al., 2014, Zanotti et al., 2015 [40,41]). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ‘Gibbs phenomenon’. In the present work, a nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion, i.e. the MOOD approach. The main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Furthermore, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Subsequently, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages.We apply the whole approach for the first time to the equations of compressible gas dynamics and magnetohydrodynamics in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme standout against a wide number of non-trivial test cases both for the compressible Navier–Stokes and the viscous and resistive magnetohydrodynamics equations. The present results show clearly that the shock-capturing capability of the news schemes is significantly enhanced within a cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS).

Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations

This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.

Residual equilibrium schemes for time dependent partial differential equations

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

Mean field games with congestion

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton–Jacobi equation and a forward Kolmogorov equation both posed in (0,T)×(RN/ZN). Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T.

Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle

The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods.

Numerical Solution of Advection–Diffusion Equation Using Meshless Method of Lines

In this paper, we present a meshless method of lines to solve one-dimensional advection–diffusion equation. For this aim, we use radial basis functions for approximate derivatives in space and fourth order Runge–Kutta scheme to solve the gained system of ordinary differential equations. Here, we use different types of radial basis functions such as multiquadric, Gaussian, inverse quadric and inverse multiquadric. The accuracy and applicability of this method are verified through the various examples. Our study shows that this method is very simple and can be easily used for solution of time-dependent partial differential equations.

Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization

We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time-dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from \[26], \[27]. It is in particular well suited for problems with time-dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinite-dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE-constrained problem with pointwise control constraints are presented.

A dynamical polynomial chaos approach for long-time evolution of SPDEs

We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen–Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier–Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

Using Newton basis functions for solving diffusion equation with nonlocal boundary condition

In this paper we introduce a meshless method based on spatial trial spaces spanned by the Newton basis functions. The Newton basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. This method can overcome on some of challenges of the full meshless kernel-based methods such as ill-conditioning associated with the dense linear systems that arise. This approach is used to obtain the numerical solution of time-dependent partial differential equations (PDEs), diffusion equation with two types of nonlocal boundary condition. The time-dependent PDE is discretized in space by the collocation method based on Newton basis functions as trial functions and it is converted to a system of ordinary differential equations in time. Stability of the scheme is discussed. Numerical examples are provided to show the accuracy and easy implementation of the novel proposed method.

Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams

This study aims at numerically analyzing the nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams with various end conditions subjected to a harmonic transverse load. Introducing a generalized displacement field including various beam theories, employing Hamilton’s principle and taking into account geometrical nonlinearity and initial imperfection, three nonlinear coupled equations and associated boundary expressions are obtained for geometrically imperfect FG-CNTRC beams. These equations formulate the longitudinal, transverse and rotational motions of FG-CNTRC beams. An efficient multistep numerical solution approach based on the generalized differential quadrature (GDQ) method, a numerical Galerkin-based scheme and time periodic discretization is employed to convert the time-dependent nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear set of ordinary differential equations (ODEs) which can be solved via the pseudo arc-length continuation technique. Nonlinear resonant dynamics characteristics are illustrated in the form of frequency-response and force-response curves; highlighting the influences of initial geometrical imperfection, geometrical parameters, excitation frequency and boundary conditions.

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

In this article, one-step semi-Lagrangian method is investigated for computing the numerical solutions of the first-order time-dependent nonlinear partial differential equations in 1D with initial and boundary conditions. This method is based on Lagrangian trajectory or the integration from the departure points to the arrival points (regular nodes) and Runge–Kutta method for ordinary differential equations. The departure points are traced back from the arrival points along the trajectory of the path. The convergence and stability are studied for the implicit and explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.

Nonlinear behaviors in a PDE model for parity-time-symmetric lasers

We propose a phenomenological time-dependent partial differential equation model to investigate the dynamical behaviors of certain parity-time () symmetric lasers during the nonlinear stage of their operations. This model incorporates physical effects such as the refractive index distribution, dispersion, material loss, nonlinear gain saturation and self-phase modulation. We show that when the loss is weak, multiple stable steady states and time-periodic states of light exist above the lasing threshold, rendering the laser multi-mode. However, when the loss is strong, only a single stable steady state of broken symmetry exists for a wide range of the gain amplitude, rendering the laser single-mode. These theoretical results corroborate the previous experimental results, and reveal the important role the loss plays in maintaining the single-mode operation of lasers.

Error analysis of sinc-Galerkin method for time-dependent partial differential equations

In this paper, we apply the Galerkin method with sinc bases for solving the boundary-value problems involving nonhomogeneous heat, convection diffusion, wave, and telegraph equations. The accuracy of the method for equation is $ \mathrm {O} (({\Delta } t)+e^{-k\sqrt {N} })$ . Error analysis is included. Several examples are given to illustrate the efficiency and implementation of the sinc-Galerkin method. Comparisons are made to confirm the reliability of the method.

Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting

Gene expression is inherently stochastic. Advanced single-cell microscopy techniques together with mathematical models for single gene expression led to important insights in elucidating the sources of intrinsic noise in prokaryotic and eukaryotic cells. In addition to the finite size effects due to low copy numbers, translational bursting is a dominant source of stochasticity in cell scenarios involving few short lived mRNA transcripts with high translational efficiency (as is typically the case for prokaryotes), causing protein synthesis to occur in random bursts.In the context of gene regulation cascades, the Chemical Master Equation (CME) governing gene expression has in general no closed form solution, and the accurate stochastic simulation of the dynamics of complex gene regulatory networks is a major computational challenge. The CME associated to a single gene self regulatory motif has been previously approximated by a one dimensional time dependent partial integral differential equation (PIDE). However, to the best of our knowledge, multidimensional versions for such PIDE have not been developed yet. Here we propose a multidimensional PIDE model for regulatory networks involving multiple genes with self and cross regulations (in which genes can be regulated by different transcription factors) derived as the continuous counterpart of a CME with jump process. The model offers a reliable description of systems with translational bursting. In order to provide an efficient numerical solution, we develop a semilagrangian method to discretize the differential part of the PIDE, combined with a composed trapezoidal quadrature formula to approximate the integral term.We apply the model and numerical method to study sustained stochastic oscillations and the development of competence, a particular case of transient differentiation attained by certain bacterial cells under stress conditions. We found that the resulting probability distributions are distinguishable from those characteristic of other transient differentiation processes. In this way, they can be employed as markers or signatures that identify such phenomena from bacterial population experimental data, for instance. The computational efficiency of the semilagrangian method makes it suitable for purposes like model identification and parameter estimation from experimental data or, in combination with optimization routines, the design of gene regulatory networks under molecular noise.

Superconvergence of Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper, we present a unified approach to study superconvergence behavior of the local discontinuous Galerkin (LDG) method for high-order time-dependent partial differential equations. We select the third and fourth order equations as our models to demonstrate this approach and the main idea. Superconvergence results for the solution itself and the auxiliary variables are established. To be more precise, we first prove that, for any polynomial of degree k, the errors of numerical fluxes at nodes and for the cell averages are superconvergent under some suitable initial discretization, with an order of $$O(h^{2k+1})$$ . We then prove that the LDG solution is $$(k+2)$$ -th order superconvergent towards a particular projection of the exact solution and the auxiliary variables. As byproducts, we obtain a $$(k+1)$$ -th and $$(k+2)$$ -th order superconvergence rate for the derivative and function value approximation separately at a class of Radau points. Moreover, for the auxiliary variables, we, for the first time, prove that the convergence rate of the derivative error at the interior Radau points can reach as high as $$k+2$$ . Numerical experiments demonstrate that most of our error estimates are optimal, i.e., the error bounds are sharp.

Solution of PDEs for First-Order Photobleaching Kinetics using Krylov Subspace Spectral Methods

We solve the first order reaction-diffusion equations which describes binding-diffusion kinetics using photobleaching scanning profile of confocal laser scanning Microscopes approximated by a Gaussian laser profile. We show how to solve the first order photobleaching kinetics partial differential equations with prebleach steady state initial conditions using a time-domain method known as a Krylov Subspace Spectral method (KSS method). KSS are explicit methods for solving time-dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of its high resolution and its superior scalability. We will apply Gaussian Quadrature rules in the spectral domain developed by Golub and Meurant to solve PDEs. We present a simple rough analytical solution, as well as a computational solution that is first-order accurate. We then use this solution to examine short and long time behaviors.

High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.