Numerical approximation of multiplicative SPDEs

We prove convergence of the stochastic exponential time differencing scheme for parabolic stochastic partial differential equations (SPDEs) with one-dimensional multiplicative noise. We examine convergence for fourth-order SPDEs and consider as an example the Swift–Hohenberg equation. After examining convergence, we present preliminary evidence of a shift in the deterministic pinning region [J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E. 73 (2006), pp. 056211-1–15; J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift–Hohenberg equation, Phys. Lett. A 360 (2007), pp. 681–688; Y.-P. Ma, J. Burke, and E. Knobloch, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1867–1883; S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1581–1592] with space–time white noise.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise

We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is almost sure uniform convergence, i.e. pathwise convergence. The proofs are based on a recent perturbation result for such equations.     

On the inclusion of the recombination term in discretizations of the semiconductor device equations

In this paper we consider the semiconductor device equations for stationary problems where the recombination term cannot be neglected. We illustrate our ideas by deriving systematically discretizations of the two continuity equations and of the Poisson equation in the case of one space dimension. This approach leads to finite difference schemes for the continuity equations which are related to the Scharfetter-Gummel scheme. For the Poisson equation we obtain a new finite difference scheme which reduces to a scheme of Mock if the mesh is uniform and the recombination is zero.     

A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics

The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from \(O(N^{3})\) required by traditional methods to O(Nlog\(^{2}N)\) and the memory requirement from \(O(N^{2})\) to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.     

Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955–987, 2016. doi: 10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler–Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.     

Queueing system with finite memory and jump intensity of the arrival process

We consider a single-server queueing system (QS) with finite memory and exponential service with parameter µ. The QS is fed by a doubly stochastic Poisson flow whose intensity λ(t) is a jump process. We propose a matrix and a functional-analytical method to study stationary and nonstationary characteristics of the QS; we also prove existence and uniqueness of a stationary regime and stabilization of a nonstationary regime of the QS. Results of numerical analysis are discussed.     

Weak Convergence of Finite Element Method for Stochastic Elastic Equation Driven By Additive Noise

In this paper we study the weak convergence of the semidiscrete and full discrete finite element methods for the stochastic elastic equation driven by additive noise, based on $C^0$ or $C^1$ piecewise polynomials. In order to simplify the analysis of weak convergence, we rewrite the stochastic elastic equation in an abstract problem and the solutions of the semidiscrete and full discrete problems in a unified form. We obtain that the weak order is twice the strong order, both in time and in space. Numerical experiments are carried out to verify the theoretical results.     

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singular nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite difference or finite volume approximations in space and time for example, is smoother, we first discretize the equation by a smooth finite difference or finite volume scheme, and then use the gPC-SG approximation to the discrete system. The jump condition at the interface is treated using the immersed upwind methods introduced in Jin (Proc Symp Appl Math 67(1):93–104, 2009) and Jin and Wen (Commun Math Sci 3:285–315, 2005). This yields a method that converges with the spectral accuracy for finite mesh size and time step. We use a linear hyperbolic equation with discontinuous and random coefficient, and the Liouville equation with discontinuous and random potential, to illustrate our idea, with both one and second order spatial discretizations. Spectral convergence is established for the first equation, and numerical examples for both equations show the desired accuracy of the method.     

A numerical simulation for the blow-up of semi-linear diffusion equations

Many mathematical models have the property of developing singularities at a finite time; in particular, the solution u(x, t) of the semi-linear parabolic Equation (1) may blow up at a finite time T. In this paper, we consider the numerical solution with blow-up. We discretize the space variables with a spectral method and the discrete method used to advance in time is an exponential time differencing scheme. This numerical simulation confirms the theoretical results of Herrero and Velzquez [M.A. Herrero and J.J.L. Velzquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), pp. 131–189.] in the one-dimensional problem. Later, we use this method as an experimental approach to describe the various possible asymptotic behaviours with two-space variables.     

Two-Dimensional Force-Free Relaxation and Superdiffusion Governed by Fractional Kinetic Equations

Starting from the integral Chapman-Kolmogorov equation and the Langevin equations with a multivariate Levy noise, we consider possible generalization of a one-dimensional fractional kinetic equations. It is assumed that the Levy noise has independent components. As the result, we get fractional Fokker-Planck equation for the probability density function in a phase space and fractional Einstein-Smoluchowski equation for the probability density function in a real space. We consider force-free superdiffusion and relaxation described by fractional equations and carry out numerical simulation of these processes by solving the corresponding Langevin equations. Analytical and numerical results agree quantitatively.     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.     

DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. A uniform numerical stability with respect to the Knudsen number \(\epsilon \), and a uniform in \(\epsilon \) error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. We provide a rigorous proof of the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.     

Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p=4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et al. (Numer. Math. 116:317–356, 2010), we add a vanishing numerical viscosity term to a semi-discrete scheme so as to preserve similar properties of stability, and thus able to prove the convergence in L ⁴-strong. The semi-discretization of the spatial structure via central finite difference method yields a stiff system of ODE. Hence, for the temporal discretization, we resort to the two-stage implicit Runge-Kutta scheme of the Gauss-Legendre type. The resulting system is unconditionally stable and possesses favorable nonlinear properties. On the other hand, despite the formation of blow up for the critical case of GKdV, it is known that a localized damping term added to the GKdV-4 equation leads to the exponential decay of the energy for small enough initial conditions, which is interesting from the standpoint of the Control Theory. Then, combining the result of convergence in L ⁴-strong with discrete multipliers and a contradiction argument, we show that the presence of the vanishing numerical viscosity term allows the uniform (with respect to the mesh size) exponential decay of the total energy associated to the semi-discrete scheme of higher-order in space with the localized damping term. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.     

Blow-up of solutions for semilinear stochastic delayed reaction–diffusion equations with Lévy noise

This paper is concerned with a class of semilinear stochastic delayed reaction–diffusion equations driven by Lévy noise in a separable Hilbert space. We establish sufficient conditions to ensure the existence of a unique positive solution. Moreover, we study blow-up of solutions in finite time in mean $L^p$-norm sense. Several examples are given to illustrate applications of the theory.     

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re \((s)\ge 0\), two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at \(s=0\). We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.     

Upwinding of the source term at interfaces for Euler equations with high friction

We consider Euler equations with a friction term that describe an isentropic gas flow in a porous domain. More precisely, we consider the transition between low and high friction regions. In the high friction region the system is reduced to a parabolic equation, the porous media equation. In this paper we present a hyperbolic approach based on a finite volume technique to compute numerical solutions for the system in both regimes. The Upwind Source at Interfaces (USI) scheme that we propose satisfies the following properties. Firstly it preserves the nonnegativity of gas density. Secondly, and this is the motivation, the scheme is asymptotically consistent with the limit model (porous media equation) when the friction coefficient goes to infinity. We show analytically and through numerical results that the above properties are satisfied. We shall also compare results given with the use of USI, hyperbolic–parabolic coupling and classical centered sources schemes.