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Gianluca Frasca-Caccia

Publications and source records attributed to Gianluca Frasca-Caccia.

11 recordsLinked to original sources
arXiv2026

Fast solution of a phase-field model of pitting corrosion

Excessive computational times represent a major challenge in the solution of corrosion models, limiting their practical applicability, e.g., as a support to predictive maintenance. In this paper, we propose an efficient strategy for solving a phase-field model for metal corrosion. Based on the Kronecker structure of the diffusion matrix in classical finite difference approximations on rectangular domains, time-stepping IMEX methods are efficiently solved in matrix form. However, when the domain is non-rectangular, the lack of the Kronecker structure prevents the direct use of the matrix-based approach. To address this issue, we reformulate the problem on an extended rectangular domain and introduce suitable iterative IMEX methods. The convergence of the iterations and the propagation of the numerical errors are analyzed. Test cases on two and three dimensional domains show that the proposed approach achieves accuracy comparable to existing methods, while significantly reducing the computational time, to the point of allowing actual predictions on standard workstations.

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arXiv2022

A new framework for polynomial approximation to differential equations

In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is further extended to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

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arXiv2022

Numerical conservation laws of time fractional diffusion PDEs

The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have conservation laws that approximate the continuous ones. In the second part of the paper, we propose a method that combines a finite difference method in space with a spectral integrator in time. The time integrator has already been applied in literature to solve time fractional equations with Caputo fractional derivative of order $α\in(0,1)$. It is here generalised to approximate Caputo and Riemann-Liouville fractional derivatives of arbitrary order. We apply the method to subdiffusion and superdiffusion equations with Riemann-Liouville fractional derivative and derive its conservation laws. Finally, we present a range of numerical experiments to show the convergence of the method and its conservation properties.

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arXiv2022

Exponentially fitted methods with a local energy conservation law

A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schroedinger equation, the new scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave.

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arXiv2021

Exponentially fitted methods that preserve conservation laws

The exponential fitting technique uses information on the expected behaviour of the solution of a differential problem to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In this cases, compared to standard methods, they have proved to be very accurate even using large integration steps. In this paper we consider exponentially fitted Runge-Kutta methods and we give characterizations of those that preserve local conservation laws of linear and quadratic quantities. As benchmark problems we consider wave equations arising as models in several fields such as fluid dynamics and quantum physics, and derive exponentially fitted methods that preserve their conservation laws of mass (or charge) and momentum. The proposed methods are applied to approximate breather wave solutions and are compared to other known methods of the same order.

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arXiv2021

Numerical preservation of multiple local conservation laws

There are several well-established approaches to constructing finite difference schemes that preserve global invariants of a given partial differential equation. However, few of these methods preserve more than one conservation law locally. A recently-introduced strategy uses symbolic algebra to construct finite difference schemes that preserve several local conservation laws of a given scalar PDE in Kovalevskaya form. In this paper, we adapt the new strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs. The Benjamin-Bona-Mahony equation and a system equivalent to the nonlinear Schroedinger equation are used as benchmarks, showing that the strategy yields conservative schemes which are robust and highly accurate compared to others in the literature.

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arXiv2021

Optimal parameters for numerical solvers of PDEs

In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.

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arXiv2019

Locally conservative finite difference schemes for the modified KdV equation

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and coworkers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.

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arXiv2019

Line Integral solution of Hamiltonian PDEs

In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg-de Vries equation, to illustrate the main features of this novel approach.

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arXiv2018

Simple bespoke preservation of two conservation laws

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic-numeric approach feasible. To illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared to others in the literature.

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