100 research outputs found
Multistage DPG time-marching scheme for nonlinear problems
In this article, we employ the construction of the time-marching
Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to
derive high-order multistage DPG methods for non-linear systems of ordinary
differential equations. The methodology extends to abstract evolution equations
in Banach spaces, including a class of nonlinear partial differential
equations. We present three nested multistage methods: the hybrid Euler method
and the two- and three-stage DPG methods. We employ a linearization of the
problem as in exponential Rosenbrock methods, so we need to compute exponential
actions of the Jacobian that change from time steps. The key point of our
construction is that one of the stages can be post-processed from another
without an extra exponential step. Therefore, the class of methods we introduce
is computationally cheaper than the classical exponential Rosenbrock methods.
We provide a full convergence proof to show that the methods are second, third,
and fourth-order accurate, respectively. We test the convergence in time of our
methods on a 2D + time semi-linear partial differential equation after a
semidiscretization in space
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
High order linearly implicit methods for semilinear evolution PDEs
This paper considers the numerical integration of semilinear evolution PDEs
using the high order linearly implicit methods developped in a previous paper
in the ODE setting. These methods use a collocation Runge--Kutta method as a
basis, and additional variables that are updated explicitly and make the
implicit part of the collocation Runge--Kutta method only linearly implicit. In
this paper, we introduce several notions of stability for the underlying
Runge--Kutta methods as well as for the explicit step on the additional
variables necessary to fit the context of evolution PDE. We prove a main
theorem about the high order of convergence of these linearly implicit methods
in this PDE setting, using the stability hypotheses introduced before. We use
nonlinear Schr\''odinger equations and heat equations as main examples but our
results extend beyond these two classes of evolution PDEs. We illustrate our
main result numerically in dimensions 1 and 2, and we compare the efficiency of
the linearly implicit methods with other methods from the litterature. We also
illustrate numerically the necessity of the stability conditions of our main
result
Exploiting Kronecker structure in exponential integrators: Fast approximation of the action ofâ phi-functions of matrices via quadrature
In this article, we propose an algorithm for approximating the action ofâ -functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of theâ -functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art
Differential Models, Numerical Simulations and Applications
This Special Issue includes 12 high-quality articles containing original research findings in the fields of differential and integro-differential models, numerical methods and efficient algorithms for parameter estimation in inverse problems, with applications to biology, biomedicine, land degradation, traffic flows problems, and manufacturing systems
On dynamical low-rank integrators for matrix differential equations
This thesis is concerned with dynamical low-rank integrators for matrix differential equations, typically stemming from space discretizations of partial differential equations. We first construct and analyze a dynamical low-rank integrator for second-order matrix differential equations, which is based on a Strang splitting and the projector-splitting integrator, a dynamical low-rank integrator for first-order matrix
differential equations proposed by Lubich and Osedelets in 2014. For the analysis, we derive coupled recursive inequalities, where we express the global error of the scheme in terms of a time-discretization error and a low-rank error contribution. The first can be treated with Taylor series expansion of the exact solution. For the latter, we make use of an induction argument and the convergence result derived by Kieri, Lubich, and Walach in 2016 for the projector-splitting integrator.
From the original method, several variants are derived which are tailored to, e.g., stiff or highly oscillatory second-order problems. After discussing details on the implementation of dynamical low-rank schemes, we turn towards rank-adaptivity. For the projector-splitting integrator we derive both a technique to realize changes in the approximation ranks efficiently and a heuristic to choose the rank appropriately over time. The core idea is to determine the rank such that the error of the low-rank
approximation does not spoil the time-discretization error. Based on the rank-adaptive pendant of the projector-splitting integrator, rank-adaptive dynamical low-rank integrators for (stiff and non-stiff) first-order and second-order matrix differential equations are derived. The thesis is concluded with numerical experiments to confirm our theoretical findings
Integration factor combined with level set method for reaction-diffusion systems with free boundary in high spatial dimensions
For reaction-diffusion equations in irregular domain with moving boundaries,
the numerical stability constraints from the reaction and diffusion terms often
require very restricted time step size, while complex geometries may lead to
difficulties in accuracy when discretizing the high-order derivatives on grid
points near the boundary. It is very challenging to design numerical methods
that can efficiently and accurately handle both difficulties. Applying an
implicit scheme may be able to remove the stability constraints on the time
step, however, it usually requires solving a large global system of nonlinear
equations for each time step, and the computational cost could be significant.
Integration factor (IF) or exponential differencing time (ETD) methods are one
of the popular methods for temporal partial differential equations (PDEs) among
many other methods. In our paper, we couple ETD methods with an embedded
boundary method to solve a system of reaction-diffusion equations with complex
geometries. In particular, we rewrite all ETD schemes into a linear combination
of specific {\phi}-functions and apply one start-of-the-art algorithm to
compute the matrix-vector multiplications, which offers significant
computational advantages with adaptive Krylov subspaces. In addition, we extend
this method by incorporating the level set method to solve the free boundary
problem. The accuracy, stability, and efficiency of the developed method are
demonstrated by numerical examples.Comment: 20 pages, 6 figures, 2 table
Multi-level stochastic collocation methods for parabolic and Schrödinger equations
In this thesis, we propose, analyse and implement numerical methods for time-dependent non-linear parabolic and Schrödinger-type equations with uncertain parameters. The discretisation of the parameter space which incorporates the uncertainty of the problem is performed via single- and multi-level collocation strategies. To deal with the possibly large dimension of the parameter space, sparse grid collocation techniques are used to alleviate the curse of dimensionality to a certain extent. We prove that the multi-level method is capable of reducing the overall computational costs significantly.
In the parabolic case, the time discretisation is performed via an implicit-explicit splitting strategy of order two which consists shortly speaking of a combination of an implicit trapezoidal rule for the stiff linear part and Heun\u27s method for the non-linear part. In the Schrödinger case, time is discretised via the famous second-order Strang splitting method.
For both problem classes we review known error bounds for both discretizations and prove new error bounds for the time discretisations which take the regularity in the parameter space into account. In the parabolic case, a new error bound for the "implicit-explicit trapezoidal method" (IMEXT) method is proved. To our knowledge, this error bound stating second-order convergence of the IMEXT method closes a current gap in the literature.
Utilising the aforementioned new error bounds for both problem classes, we can rigorously prove convergence of the single- and multi-level methods. Additionally, cost savings of the multi-level methods compared to the single-level approach are predicted and verifed by numerical examples.
The results mentioned above are novel contributions in two areas of mathematics. The first one is (analysis of) numerical methods for uncertainty quantification and the second one is numerical analysis of time-integration schemes for PDEs
On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation
Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of , where is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six
Positivity-preserving methods for ordinary differential equations
[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056
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