Discrete Numerical Approximation Research Articles

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

Strong convergence rates for fuly discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities, such as the stochastic Allen–Cahn equation with space-time white noise, are shown. The obtained strong rates of convergence are essentially sharp.

Numerical Approximation of the Value of a Stochastic Differential Game with Asymmetric Information

We consider a convexity constrained Hamilton-Jacobi-Bellman-type obstacle problem for the value function of a zero-sum differential game with asymmetric information. We propose a convexity-preserving probabilistic numerical scheme for the approximation of the value function which is discrete w.r.t. the time and convexity variables, and show that the scheme converges to the unique viscosity solution of the considered problem. Furthermore, we generalize the semi-discrete scheme to obtain an implementable fully discrete numerical approximation of the value function and present numerical experiments to demonstrate the properties of the proposed numerical scheme.

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have recently been obtained in [Becker, S., Gess, B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. arXiv:1711.02423 (2017)].

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.

Exponential moments for numerical approximations of stochastic partial differential equations

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish positive strong convergence rates for the considered approximation scheme. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and two-dimensional stochastic Navier-Stokes equations.

Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems

Peridynamics is widely used as the theoretical basis for numerical studies of fracture evolution, propagation, and behavior. While the theory has been shown to converge to continuum mechanics in the theoretical limit, its behavior as a discrete numerical approximation with respect to classic problems has not been shown. In this study, we use standard analytical solutions to thoroughly test the numerical accuracy and rate of convergence of the spatial discretization obtained by peridynamics. We analyze the accuracy and rate of convergence of three different peridynamic constitutive responses: of these, two involve a state-based dilation, and the third is based on the estimation of the deformation gradient. Additionally, we study the choice of the peridynamic influence function in each of the constitutive responses. The peridynamic materials are solved in the linear elastic regime by solving a linear system of equations obtained by symbolically differentiating the force states. We test the methods against standard constant-strain solutions for uniaxial compression, isotropic compression, and simple shear. We also apply the methods to a finite material with a pressurized thin crack, using the Westergaard’s solution method to obtain an analytical displacement field for comparison. The two dilation-based peridynamic constitutive responses are found to only converge to one of the constant strain solutions, while the deformation gradient-based law converges in all cases with an appropriate choice of the influence functions. We show that a cubic influence function is the best choice of those considered in all methods. Only the deformation gradient-based model converges for all three linear deformation problems, but is less accurate than the dilation-based models for the thin crack problem because of instabilities. We demonstrate an ad hoc smoothing technique based on the influence function that is able to alleviate these instabilities and improve the accuracy of the deformation gradient-based model.

Discrete port-Hamiltonian formulation and numerical approximation for systems of two conservation laws

Abstract We discuss the discrete formulation of systems of conservation laws in port-Hamiltonian form on dual chain complexes. Based on integral balance equations and topological information, this representation is exact and qualifies as a control model. The finite-dimensional approximation requires an energy discretization that yields discrete constitutive equations. We give (i) a brief overview of discrete modeling of conservation laws on n-complexes and (ii) extend existing results by allowing for mixed physical types of boundary inputs. This requires the construction of a primal and a dual complex based on the underlying staggered grids and the localization of the inputs on the system boundary. Finally, (iii) we discuss the properties of the resulting structure-preserving discretization scheme based on a consistency analysis for the 2D nonlinear shallow water equations.

Fractional difference/finite element approximations for the time–space fractional telegraph equation

In this paper, we study the numerical solution of the time–space fractional order (fractional for simplicity) telegraph equation, which can be used in signal analysis for transmission and propagation of electrical signals, also the modeling of the reaction diffusion and the random walk of suspension flows and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the spatial Riemann–Liouville fractional derivative with order 1+β∈(1,2) and the finite difference schemes for the temporal Caputo derivatives with orders α∈(1/2,1) and 2α are analyzed respectively. Results on the existence and uniqueness of the solution, the numerical stability, and the error estimates are displayed in details. Numerical examples are included to confirm the theoretical analysis.

REGULARIZATION SCHEMES FOR DEGENERATE RICHARDS EQUATIONS AND OUTFLOW CONDITIONS

We analyze regularization schemes for the Richards equation and a time discrete numerical approximation. The original equations can be doubly degenerate, therefore they may exhibit fast and slow diffusion. In addition, we treat outflow conditions that model an interface separating the porous medium from a free flow domain. In both situations we provide a regularization with a non-degenerate equation and standard boundary conditions, and discuss the convergence rates of the approximations.

Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion

In this paper, we study the time–space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann–Liouville fractional derivative with order 1 + β ∈ [ 1 , 2 ] and the finite difference scheme for the time Caputo derivative with order α ∈ ( 0 , 1 ) (for subdiffusion) and ( 1 , 2 ) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0 < α < 1 is slower than that for α = 1 , but the diffusion velocity for 1 < α < 2 is faster than that for α = 1 . For the spatial diffusion, we have a similar observation.

Numerical Approximation of a Time Dependent, Nonlinear, Space‐Fractional Diffusion Equation

In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity. The analysis is performed for a general fractional order diffusion operator. The nonlinear term studied is a product of the unknown function and a convolution operator of order 0. Convergence of the approximation and a priori error estimates are given. Numerical computations are included, which confirm the theoretical predictions.

Statistically relevant conserved quantities for truncated quasigeostrophic flow.

Systematic applications of ideas from equilibrium statistical mechanics lead to promising strategies for assessing the unresolved scales of motion in many problems in science and engineering. A scientific debate over more than the last 25 years involves which conserved quantities among the formally infinite list are statistically relevant for the large-scale equilibrium statistical behavior. Here this important issue is addressed by using suitable discrete numerical approximations for geophysical flows with many conserved quantities as a numerical laboratory. The results of numerical experiments are presented here for these truncated geophysical flows with topography in a suitable regime. These experiments establish that the integrated third power of potential vorticity besides the familiar constraints of energy, circulation, and enstrophy (the integrated second power) is statistically relevant in this regime for the coarse-grained equilibrium statistical behavior at large scales. Furthermore, the integrated higher powers of potential vorticity larger than three are statistically irrelevant for the large-scale equilibrium statistical behavior in the examples studied here.

Discrete Approximations with Additional Conserved Quantities: Deterministic and Statistical Behavior

Discrete numerical approximations with additional conserved quantities are devel- oped here both for barotropic geophysical flows generalizing the 2D incompressible fluid equations and suitable discretizations of the Burgers-Hopf equation. Mathematical, numerical, and statistical properties of these approximations are studied below in various different contexts through the sym- biotic interaction of mathematical theory and scientific computing. The new contributions include an explicit concrete discussion of the sine-bracket spectral truncation with many conserved quanti- ties for 2D incompressible flow, a theoretical and numerical comparison with the standard spectral truncation, and a rigorous proof of convergence to suitable weak solutions in the limit as the number of Fourier modes increases. Systematic discretizations of the Burgers-Hopf equation are developed, which conserve linear momentum and a non-linear energy; careful numerical experiments regarding the statistical behavior of these models indicate that they are ergodic and strongly mixing but do not have equipartition of energy in Fourier space. Furthermore, the probability distribution of the values at a single grid point can be highly non-Gaussian with two strong isolated peaks in this distribution. This contrasts with earlier results for statistical behavior of difference schemes which conserve a quadratic energy. The issues of statistically relevant conserved quantities are introduced through a new case study for the Galerkin-truncated Burgers-Hopf model.

Approximation of Time-Dependent Viscoelastic Fluid Flow: SUPG Approximation

In this article we consider the numerical approximation to the time-dependent viscoelasticity equations with an Oldroyd B constitutive equation. The approximation is stabilized by using a streamline upwind Petrov--Galerkin (SUPG) approximation for the constitutive equation. We analyze both the semidiscrete and fully discrete numerical approximations. For both discretizations we prove the existence of, and derive a priori error estimates for, the numerical approximations.

A Dynamic Frictional Contact Problem with Normal Damped Response

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a reactive foundation. The process is assumed to be dynamic and the contact is modeled with a general normal damped response condition and a local friction law. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using results on evolution equations with monotone operators and a fixed point argument. We then introduce and study a fully discrete numerical approximation scheme of the variational problem, in terms of the velocity variable. The numerical scheme has a unique solution. We derive error estimates under additional regularity assumptions on the data and the solution.

Pricing Complex Barrier Options under General Diffusion Processes

Many derivations have payoffs that depend on whether a prespecified price “barrier” is hit prior to option maturity. Standard “in” and “out” options are obvious examples, but more complex contingencies, like a debt coverage ratio covenant in a bond indenture, may also be modeled as barrier problems. Such options present serious challenges to numerical valuation techniques, because accuracy depends heavily on where the barrier falls relative to the nodes in a discrete numerical approximation lattice. For good performance, the barrier should coincide with a set of nodes. Standard models typically perform well only for a single barrier of a particular shape, such as an “outstrike” at a constant price. Non-constant volatility for the underlying asset is an insuperable problem, leading to very slow convergence in most models. Here, Tian shows how to restructure a trinomial lattice to allow time-varying volatility, while maintaining the ability to match both an upper and a lower barrier of general form. The result is an accurate numerical approximation that converges very rapidly.

Fully discrete approximation methods for the estimation of parabolic systems and boundary parameters

A spatially and temporally discrete numerical approximation scheme is developed for the identification of a class of semilinear parabolic systems with unknown boundary parameters. The identification problem is formulated as a least squares fit to data subject to an equivalent representation for the dynamics in the form of an abstract evolution equation. Finite-dimensional difference equation state approximations are constructed using a cubic spline-based, Galerkin method and the Pade rational function approximations to the exponential. A sequence of approximating identification problems result, the solutions of which are shown to exist and, in a certain sense, approximate solutions to the original identification problem. Numerical results for two examples, one involving the modeling of biological mixing in deep sea sediment cores, and the other, the estimation of transport parameters for indoor mixing, are discussed. In both examples, the identification is based upon actual experimental data.