Galerkin Discretization In Space Research Articles

Reduced-Order Modeling for Heston Stochastic Volatility Model

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Space-Time Discontinuous Galerkin Methods for Weak Solutions of Hyperbolic Linear Symmetric Friedrichs Systems

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the {textrm{L}}_2 norm at final time. Numerical experiments confirm that the a priori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator.

Semi-implicit Hybrid Discrete $$\left( \text {H}^T_N\right) $$ Approximation of Thermal Radiative Transfer

The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing numerical solutions of the TRT equations accurately and efficiently is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P\(_N\) (spherical harmonics) or the S\(_N\) (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H\(^T_N\)) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P\(_N\) and S\(_N\), and indeed reduces to each of these approximations in various limits: H\(^1_N\) \(\equiv \) P\(_N\) and H\(^T_0\) \(\equiv \) S\(_T\). We prove that H\(^T_N\) results in a system of hyperbolic partial differential equations for all \(T\ge 1\) and \(N\ge 0\). Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop an asymptotic-preserving numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge–Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H\(^T_N\) ansatz and the numerical discretizations.

A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an hp-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in - and -type norms when I is the temporal and the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. We also show how our approach allows the derivation of such bounds in the norm.

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.

An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations

A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy-preserving integrator. The relay leads to a fully discretized and energy-preserved scheme. This strategy is fully realized for solving a nonlinear Schrodinger equation through a combination of the Galerkin discretization in space and a Crank–Nicolson scheme in time. The order of convergence of our new method is O ( τ 2 + h 2 ) if the discrete L2-norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy-preserving scheme constructed.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1485–1504, 2016

A class of Galerkin Schemes for Time-Dependent Radiative Transfer

The numerical solution of time-dependent radiative transfer problems is challenging, due to the high dimension and to the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and also to preserve basic properties such as exponential stability and decay to equilibrium on the discrete level. We present the basic a priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure to first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis f...

Implicit Runge--Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations

In this paper we consider implicit Runge--Kutta methods for the time integration of linear Maxwell's equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauss and Radau collocation methods. The results for the abstract evolution equation are then combined with a discontinuous Galerkin discretization in space using upwind fluxes. For the case that permeability and permittivity are piecewise constant functions, we show error bounds for the full discretization, where the constants do not deteriorate if the spatial mesh width tends to zero.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro‐magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first‐order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro‐magnetic and elastic waves.

Discontinuous Galerkin methods for solving a quasistatic contact problem

We consider the numerical solution of a nonlinear evolutionary variational inequality, arising in the study of quasistatic contact problems. We study spatially semi-discrete and fully discrete schemes for the problem with several discontinuous Galerkin discretizations in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for the schemes, reaching the optimal convergence order for linear elements. Numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders.

An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell's equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension $$\kappa \times \kappa $$ ? × ? . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram---Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension $$m\times m$$ m × m ( $$m\ll \kappa $$ m ? ? ). For computing the matrix exponential, we obtain eigenvalues of the $$m\times m$$ m × m matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally $$O(\Delta t^{m-1})$$ O ( Δ t m - 1 ) in time so that it allows taking a larger timestep size as $$m$$ m increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge---Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1+Δts), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.

Galerkin single-step methods for second-order hyperbolic equations

Single-step methods, coupled with Galerkin discretizations in space, are applied to second-order hyperbolic equations. These methods are applied directly to the second-order equations. Optimal order convergence estimates are derived.