Numerical methods for solving hyperbolic and parabolic partial differential equations

The main object of this thesis is a study of the numerical 'solution of hyperbolic and parabolic partial differential equations. The introductory chapter deals with a general description and classification of partial differential equations. Some useful mathematical preliminaries and properties of matrices are outlined. Chapters Two and Three are concerned with a general survey of current numerical methods to solve these equations. By employing finite differences, the differential system is replaced by a large matrix system. Important concepts such as convergence, consistency, stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah on parabolic equations are now applied to first and second order (wave equation) hyperbolic equations in Chapter 4. By coupling existing difference equations to approximate the given hyperbolic equations, new GE schemes are introduced. Their accuracies and truncation errors are studied and their stabilities established. Chapter 5 deals with the application of the GE techniques on some commonly occurring examples possessing variable coefficients such as the parabolic diffusion equations with cylindrical and spherical symmetry. A complicated stability analysis is also carried out to verify the stability, consistency and convergence of the proposed scheme. In Chapter 6 a new iterative alternating group explicit (AGE) method with the fractional splitting strategy is proposed to solve various linear and non-linear hyperbolic and parabolic problems in one dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of difference schemes and proved to be stable. Its rate of convergence is governed by the acceleration parameter and with an optimum choice of this parameter, it is found that the accuracy of this method, in general, is better if not comparable to that of the GE class of problems as well as other existing schemes. The work on the AGE algorithm is extended to parabolic problems of two and three space dimensions in Chapter 7. A number of examples are treated and the DR variant is used because of consideration of stability requirement. The thesis ends with a summary and recommendations for future work

Non-Gaussian Quadrature Integral Transform Solution of Parabolic Models with a Finite Degree of Randomness

[EN] In this paper, we propose an integral transform method for the numerical solution of random mean square parabolic models, that makes manageable the computational complexity due to the storage of intermediate information when one applies iterative methods. By applying the random Laplace transform method combined with the use of Monte Carlo and numerical integration of the Laplace transform inversion, an easy expression of the approximating stochastic process allows the manageable computation of the statistical moments of the approximation.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Casabán Bartual, MC.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Non-Gaussian Quadrature Integral Transform Solution of Parabolic Models with a Finite Degree of Randomness. Mathematics. 8(7):1-16. https://doi.org/10.3390/math8071112S11687Bharucha-Reid, A. T. (1964). On the theory of random equations. Proceedings of Symposia in Applied Mathematics, 40-69. doi:10.1090/psapm/016/0189071Ernst, O. G., Sprungk, B., & Tamellini, L. (2018). Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs). SIAM Journal on Numerical Analysis, 56(2), 877-905. doi:10.1137/17m1123079Casaban, M.-C., Cortes, J.-C., & Jodar, L. (2018). Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach. Mathematical Modelling and Analysis, 23(1), 79-100. doi:10.3846/mma.2018.006Casabán, M. C., Company, R., & Jódar, L. (2019). Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness. Mathematics, 7(9), 853. doi:10.3390/math7090853Nouri, K., & Ranjbar, H. (2014). Mean Square Convergence of the Numerical Solution of Random Differential Equations. Mediterranean Journal of Mathematics, 12(3), 1123-1140. doi:10.1007/s00009-014-0452-8Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2015). A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259, 654-667. doi:10.1016/j.amc.2015.02.091Iserles, A. (2004). On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. IMA Journal of Numerical Analysis, 24(3), 365-391. doi:10.1093/imanum/24.3.365Consuelo Casabán, M., Company, R., Egorova, V. N., & Jódar, L. (2020). Integral transform solution of random coupled parabolic partial differential models. Mathematical Methods in the Applied Sciences, 43(14), 8223-8236. doi:10.1002/mma.6492Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics, 330, 937-954. doi:10.1016/j.cam.2016.11.049Davies, B., & Martin, B. (1979). Numerical inversion of the laplace transform: a survey and comparison of methods. Journal of Computational Physics, 33(1), 1-32. doi:10.1016/0021-9991(79)90025-1Ng, E. W., & Geller, M. (1969). A table of integrals of the Error functions. Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences, 73B(1), 1. doi:10.6028/jres.073b.001Armstrong, J. S., & Collopy, F. (1992). Error measures for generalizing about forecasting methods: Empirical comparisons. International Journal of Forecasting, 8(1), 69-80. doi:10.1016/0169-2070(92)90008-

Inverse Problems of Determining Sources of the Fractional Partial Differential Equations

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary condition

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Multiscale differential Riccati equations for linear quadratic regulator problems

We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the $L^2$ operator norm, and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations, and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs from the previous one only by the addition of Remark 7.2 and minor changes in formatting. 21 pages, 12 figure