Numerical methods for stochastic partial differential equations with white noise. ([2017])

• Record Type: Book
• Further Information: Note: Zhongqiang Zhang, George Em Karniadakis.
• Authors: Zhang, Zhongqiang; Karniadakis, George
• Contents: Preface; Contents; 1 Prologue; 1.1 Why random and Brownian motion (white noise)?; What color is the noise?; Solutions to (1.1.2); 1.2 Modeling with SPDEs; 1.3 Specific topics of this book; 2 Brownian motion and stochastic calculus; 2.1 Gaussian processes and their representations; 2.2 Brownian motion and white noise; 2.2.1 Some properties of Brownian motion; Regularity of Brownian motion; 2.2.2 Approximation of Brownian motion; 2.3 Brownian motion and stochastic calculus; 2.4 Stochastic chain rule: Ito formula; 2.5 Integration methods in random space; 2.5.1 Monte Carlo method and its variants. Multilevel Monte Carlo method2.5.2 Quasi-Monte Carlo methods; 2.5.3 Wiener chaos expansion method; 2.5.4 Stochastic collocation method ; Smolyak's sparse grid; 2.5.5 Application to SODEs; 2.6 Bibliographic notes; 2.7 Suggested practice; 3 Numerical methods for stochastic differential equations; 3.1 Basic aspects of SODEs; 3.1.1 Existence and uniqueness of strong solutions; 3.1.2 Solution methods; The integrating factor method; Moment equations of solutions; 3.2 Numerical methods for SODEs; 3.2.1 Derivation of numerical methods based on numerical integration; 3.2.2 Strong convergence. 3.2.3 Weak convergence3.2.4 Linear stability; 3.2.5 Summary of numerical SODEs; 3.3 Basic aspects of SPDEs; 3.3.1 Functional spaces; 3.3.2 Solutions in different senses; 3.3.3 Solutions to SPDEs in explicit form; 3.3.4 Linear stochastic advection-diffusion-reactionequations; 3.3.5 Existence and uniqueness; 3.3.6 Conversion between Ito and Stratonovichformulation; 3.4 Numerical methods for SPDEs; 3.4.1 Direct semi-discretization methods for parabolicSPDEs; Second-order equations; Fourth-order equations; 3.4.2 Wong-Zakai approximation for parabolic SPDEs. 3.4.3 Preprocessing methods for parabolic SPDEsSplitting methods; Integrating factor (exponential integrator) techniques; 3.4.4 What could go wrong? Examples of stochastic Burgers and Navier-Stokes equations; 3.4.5 Stability and convergence of existing numericalmethods; Weak convergence; Pathwise convergence; Stability; 3.4.6 Summary of numerical SPDEs; 3.5 Summary and bibliographic notes; 3.6 Suggested practice; Part I Numerical Stochastic Ordinary Differential Equations; 4 Numerical schemes for SDEs with time delay using the Wong-Zakai approximation; 4.1 Wong-Zakai approximation for SODEs. 4.1.1 Wong-Zakai approximation for SDDEs4.2 Derivation of numerical schemes; 4.2.1 A predictor-corrector scheme; 4.2.2 The midpoint scheme; 4.2.3 A Milstein-like scheme; 4.3 Linear stability of some schemes; Stability region of the forward Euler scheme; Stability analysis of the predictor-corrector scheme; Stability analysis of the midpoint scheme; 4.4 Numerical results; 4.5 Summary and bibliographic notes; 4.6 Suggested practice; 5 Balanced numerical schemes for SDEs with non-Lipschitzcoefficients; 5.1 A motivating example; 5.2 Fundamental theorem; 5.2.1 On application of Theorem 5.2.3.
• Publisher Details: Cham, Switzerland : Springer
• Publication Date: 2017
• Copyright Date: 2017
• Extent: 1 online resource
• Subjects: 519.2/2; Mathematics; Stochastic partial differential equations; MATHEMATICS -- Applied; MATHEMATICS -- Probability & Statistics -- General; Stochastic partial differential equations; Mathematics -- Differential Equations; Probability & statistics; Differential calculus & equations; Numerical analysis; Distribution (Probability theory); Differential equations, partial; Mathematics -- Number Systems; Numerical analysis; Electronic books
• Languages: English
• ISBNs: 9783319575117; 3319575112
• Related ISBNs: 9783319575100; 3319575104
• Notes: Note: Includes bibliographical references and index.; Note: Print version record.
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