Higher-order Splitting Methods Research Articles

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

A Large Scale Benchmark for the Incompressible Navier-Stokes Equations

We introduce a collection of benchmark problems in 2D and 3D (geometry description and boundary conditions), including simple cases with known analytic solution, classical experimental setups, and complex geometries with fabricated solutions for evaluation of numerical schemes for incompressible Navier-Stokes equations in laminar flow regime. We compare the performance of a representative selection of most broadly used algorithms for Navier-Stokes equations on this set of problems. Where applicable, we compare the most common spatial discretization choices (unstructured triangle/tetrahedral meshes and structured or semi-structured quadrilateral/hexahedral meshes). The study shows that while the type of spatial discretization used has a minor impact on the accuracy of the solutions, the choice of time integration method, spatial discretization order, and the choice of solving the coupled equations or reducing them to simpler subproblems have very different properties. Methods that are directly solving the original equations tend to be more accurate than splitting approaches for the same number of degrees of freedom, but numerical or computational difficulty arise when they are scaled to larger problem sizes. Low-order splitting methods are less accurate, but scale more easily to large problems, while higher-order splitting methods are accurate but require dense time discretizations to be stable. We release the description of the experiments and an implementation of our benchmark, which we believe will enable statistically significant comparisons with the state of the art as new approaches for solving the incompressible Navier-Stokes equations are introduced.

Dispersive fractalisation in linear and nonlinear Fermi–Pasta–Ulam–Tsingou lattices

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at timestwhere these models have validity, namelyt= O(h−2), wherehis proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.

Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations

In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrödinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 > ε > > 1 0 > \varepsilon >\!\!> 1 , defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor ε \varepsilon , provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.

A splitting approach for the Kadomtsev–Petviashvili equation

We consider a splitting approach for the Kadomtsev–Petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented. The error made by this numerical scheme is compared to exponential integrators which have been shown in Klein and Roidot (2011) [2] to perform best for stiff solutions of the Kadomtsev–Petviashvili equation. Since many classic high order splitting methods do not perform well, we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four. In addition, the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated.

Numerical solution of Burgers’ equation with high order splitting methods

In this work, high order splitting methods have been used for calculating the numerical solutions of Burgers’ equation in one space dimension with periodic, Dirichlet, Neumann and Robin boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and cannot be used for time-irreversible systems, such as Burgers’ equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem (or one that can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers’ equation. The numerical results show that both the methods with complex time steps having one set of coefficients real and positive, say ai∈R+ and bi∈C+, and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of Burgers’ equation.

Defect-based local error estimators for high-order splitting methods involving three linear operators

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Efficient time integration methods based on operator splitting and application to the Westervelt equation

High-order splitting methods for separable non-autonomous parabolic equations

We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time-dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.

An extrapolated splitting method for solving semi-discretized parabolic differential equations

An extrapolated Peaceman–Rachford–Strang splitting method is designed and examined in application to semi-discretized parabolic partial differential equations. A multi-product expansion method is implemented to improve the order of accuracy beyond second order in time. The numerical analysis of the high-order splitting method is further validated and illustrated through numerical experiments of linear and nonlinear partial differential equations.

Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrodinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order p applied to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations, which are associated with a model reduction for high-dimensional linear Schrodinger equations describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p − 1 in the H 1 Sobolev norm and convergence of order p in the L 2 norm is proven. An analogous result follows for the cubic nonlinear Schrodinger equation, which is also illustrated by a numerical experiment.

Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II. Higher-order methods for linear problems

In this work, defect-based local error estimators for higher-order exponential operator splitting methods are constructed and analyzed in the context of time-dependent linear Schrödinger equations. The technically involved procedure is carried out in detail for a general three-stage third-order splitting method and then extended to the higher-order case. Asymptotical correctness of the a posteriori local error estimator is proven under natural commutator bounds for the involved operators, and along the way the known (non)stiff order conditions and a priori convergence bounds are recovered. The theoretical error estimates for higher-order splitting methods are confirmed by numerical examples for a test problem of Schrödinger type. Further numerical experiments for a test problem of parabolic type complement the investigations.

Optimized high-order splitting methods for some classes of parabolic equations

We are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this se cond-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, method s of orders 3 to 14 by using the Suzuki‐Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8.

On the efficiency of quantum algorithms for Hamiltonian simulation

We study the efficiency of algorithms simulating a system evolving with Hamiltonian $H=\sum_{j=1}^m H_j$. We consider high order splitting methods that play a key role in quantum Hamiltonian simulation. We obtain upper bounds on the number of exponentials required to approximate $e^{-iHt}$ with error $\e$. Moreover, we derive the order of the splitting method that optimizes the cost of the resulting algorithm. We show significant speedups relative to previously known results.

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Holder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})$ for n≥k, where k=k(Q) is determined by topological properties of the loop and e>0 is arbitrarily small. The convergence rate is therefore e-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.

Close-to-optimal bounds for [formula omitted] loop approximation

In Oswald and Shingel (2009) [6], we proved an asymptotic O ( n − α / ( α + 1 ) ) bound for the approximation of SU ( N ) loops ( N ≥ 2 ) with Lipschitz smoothness α > 1 / 2 by polynomial loops of degree ≤ n . The proof combined factorizations of SU ( N ) loops into products of constant SU ( N ) matrices and loops of the form e A ( t ) where A ( t ) are essentially su ( 2 ) loops preserving the Lipschitz smoothness, and the careful estimation of errors induced by approximating matrix exponentials by first-order splitting methods. In the present note we show that using higher order splitting methods allows us to improve the above suboptimal result to close-to-optimal O ( n − ( α − ϵ ) ) bounds for α > 1 , where ϵ > 0 can be chosen arbitrarily small.

Splitting methods with complex times for parabolic equations

Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations. We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.