Suchergebnisse

Abstract. The initial conditions for the simulation of a seismically induced tsunami for a rapid, assumed-to-be-instantaneous vertical seafloor displacement is given by the Kajiura low-pass filter integral. This work proposes a new, efficient, and accurate approach for its numerical evaluation, valid when the seafloor displacement is discretized as a set of rectangular contributions over variable bathymetry. We compare several truncated quadrature formulae, selecting the optimal one. The reconstruction of the initial sea level perturbation as a linear combination of pre-computed elementary sea surface displacements is tested on the tsunamigenic Kuril earthquake doublet – a megathrust and an outer rise – that occurred in the central Kuril Islands in late 2006 and early 2007. We also confirm the importance of the horizontal contribution to tsunami generation, and we consider a simple model of the inelastic deformation of the wedge on realistic bathymetry. The proposed approach results are accurate and fast enough to be considered relevant for practical applications. A tool to build a tsunami source database for a specific region of interest is provided.

1 Introduction -- 1.1 What are numerical methods? -- 1.2 Numerical methods versus numerical analysis -- 1.3 Why use numerical methods? -- 1.4 Approximate equations and approximate solutions -- 1.5 The use of numerical methods -- 1.6 Errors -- 1.7 Non-dimensional equations -- 1.8 The use of computers -- 2 The solution of equations -- 2.1 Introduction -- 2.2 Location of initial estimates -- 2.3 Interval halving -- 2.4 Simple iteration -- 2.5 Convergence -- 2.6 Aitken's extrapolation -- 2.7 Damped simple iteration -- 2.8 Newton-Raphson method -- 2.9 Extended Newton's method -- 2.10 Other iterative methods -- 2.11 Polynomial equations -- 2.12 Bairstow's method 56 Worked examples 58 Problems -- 3 Simultaneous equations -- 3.1 Introduction -- 3.2 Elimination methods -- 3.3 Gaussian elimination -- 3.4 Extensions to the basic algorithm -- 3.5 Operation count for the basic algorithm -- 3.6 Tridiagonal systems -- 3.7 Extensions to the Thomas algorithm -- 3.8 Iterative methods for linear systems -- 3.9 Matrix inversion -- 3.10 The method of least squares -- 3.11 The method of differential correction -- 3.12 Simple iteration for non-linear systems -- 3.13 Newton's method for non-linear systems -- Worked examples -- Problems -- 4 Interpolation, differentiation and integration -- 4.1 Introduction -- 4.2 Finite difference operators -- 4.3 Difference tables -- 4.4 Interpolation -- 4.5 Newton's forward formula -- 4.6 Newton's backward formula -- 4.7 Stirling's central difference formula -- 4.8 Numerical differentiation -- 4.9 Truncation errors -- 4.10 Summary of differentiation formulae -- 4.11 Differentiation at non-tabular points: maxima and minima -- 4.12 Numerical integration -- 4.13 Error estimation -- 4.14 Integration using backward differences -- 4.15 Summary of integration formulae -- 4.16 Reducing the truncation error 146 Worked examples 149 Problems -- 5 Ordinary differential equations -- 5.1 Introduction -- 5.2 Euler's method -- 5.3 Solution using Taylor's series -- 5.4 The modified Euler method -- 5.5 Predictor-corrector methods -- 5.6 Milne's method, Adams' method, and Hamming's method -- 5.7 Starting procedure for predictor-corrector methods -- 5.8 Estimation of error of predictor-corrector methods -- 5.9 Runge-Kutta methods -- 5.10 Runge-Kutta-Merson method -- 5.11 Application to higher-order equations and to systems -- 5.12 Two-point boundary value problems -- 5.13 Non-linear two-point boundary value problems 198 Worked examples 199 Problems -- 6 Partial differential equations I — elliptic equations -- 6.1 Introduction -- 6.2 The approximation of elliptic equations -- 6.3 Boundary conditions -- 6.4 Non-dimensional equations again -- 6.5 Method of solution -- 6.6 The accuracy of the solution -- 6.7 Use of Richardson's extrapolation -- 6.8 Other boundary conditions -- 6.9 Relaxation by hand-calculation -- 6.10 Non-rectangular solution regions -- 6.11 Higher-order equations 238 Problems -- 7 Partial differential equations II — parabolic equations -- 7.1 Introduction -- 7.2 The conduction equation -- 7.3 Non-dimensional equations yet again -- 7.4 Notation -- 7.5 An explicit method -- 7.6 Consistency -- 7.7 The Dufort-Frankel method -- 7.8 Convergence -- 7.9 Stability -- 7.10 An unstable finite difference approximation -- 7.11 Richardson's extrapolation 261 Worked examples 262 Problems -- 8 Integral methods for the solution of boundary value problems -- 8.1 Introduction -- 8.2 Integral methods -- 8.3 Implementation of integral methods 271 Worked examples 278 Problems -- Suggestions for further reading.

"In this paper we investigate a collocation method for the approximate
solution of Hammerstein integral equations in two dimensions. As in [8], col-
location is applied to a reformulation of the equation in a new unknown, thus
reducing the computational cost and simplifying the implementation. We start
with a special type of piecewise linear interpolation over triangles for a refor-
mulation of the equation. This leads to a numerical integration scheme that can
then be extended to any bounded domain in R2, which is used in collocation. We
analyze and prove the convergence of the method and give error estimates. As
the quadrature formula has a higher degree of precision than expected with linear
interpolation, the resulting collocation method is superconvergent, thus requiring
fewer iterations for a desired accuracy. We show the applicability of the proposed
scheme on numerical examples and discuss future research ideas in this area."

The first author acknowledges partial financial support by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. The second author has been partially supported by Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) : BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and by Ramon y Cajal with reference RYC-2017-22649. The fourth author is member of GNCS-INdAM. ; Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m →∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests. ; IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033 ; Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) SEV-2017-0718 ; Spanish Government RYC-2017-22649

A numerical method for computational implementation of gradient dynamical systems is presented. The method is based upon the development of geometric integration numerical methods, which aim at preserving the dynamical properties of the original ordinary differential equation under discretization. In particular, the proposed method belongs to the class of discrete gradients methods, which substitute the gradient of the continuous equation with a discrete gradient, leading to a map that possesses the same Lyapunov function of the dynamical system, thus preserving the qualitative properties regardless of the step size. In this work, we apply a discrete gradient method to the implementation of Hopfield neural networks. Contrary to most geometric integration methods, the proposed algorithm can be rewritten in explicit form, which considerably improves its performance and stability. Simulation results show that the preservation of the Lyapunov function leads to an improved performance, compared to the conventional discretization. ; Spanish Government project no. TIN2010-16556 Junta de Andalucía project no. P08-TIC-04026 Agencia Española de Cooperación Internacional para el Desarrollo project no. A2/038418/11