Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

Numerical solution of systems of partial integral differential equations with application to pricing options

We introduce and analyze a strongly stable numerical method designed to yield good performance under challenging conditions of irregular or mismatched initial data for solving systems of coupled partial integral differential equations (PIDEs). Spatial derivatives are approximated using second order central difference approximations by treating the mixed derivative terms in a special way. The integral operators are approximated using one and two–dimensional trapezoidal rule on an equidistant grid. Computational complexity of the method for solving large systems of PIDEs is discussed. A detailed treatment for the consistency, stability, and convergence of the proposed method is provided. Two asset American option under regime–switching with jump–diffusion model when solved using a penalty term, leads to a system of two dimensional PIDEs with mixed derivatives. This model involves double probability density function which brings more challenges to the numerical solution in already a complicated partial integral differential equation. The complexity of the dense jump probability generator, the nonlinear penalty term and the regime–switching terms are treated efficiently, while maintaining the stability and convergence of the method. The impact of the jump intensity and other parameters is shown in the graphs. Numerical experiments are performed to demonstrated efficiency, accuracy, and reliability of the proposed approach.     

Solution of lambda-omega systems: Theta-schemes and multigrid methods

Summary. The numerical solution of a time-dependent reaction diffusion lambda-omega system discretized by theta-schemes and finite differences is considered. Stability and accuracy of finite difference theta-schemes for this system are established. To solve the time-implicit evolution equations a nonlinear multigrid method is applied. The convergence properties of this solver are investigated considering a linearized lambda-omega model.Mathematics Subject Classification (1991): 35K57, 65M06, 65M12, 65M55, 65P20, 65P40Revised version received February 3, 2004     

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L_∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization. AMS subject classification 65L10, 65L12, 65L15     

Convergence analysis of the local defect correction method for diffusion equations

Summary. This paper is concerned with the convergence analysis of the local defect correction (LDC) method for diffusion equations. We derive a general expression for the iteration matrix of the method. We consider the model problem of Poisson's equation on the unit square and use standard five-point finite difference discretizations on uniform grids. It is shown via both an upper bound for the norm of the iteration matrix and numerical experiments, that the rate of convergence of the LDC method is proportional to H ² with H the grid size of the global coarse grid. Mathematics Subject Classification (2000):65N22, 65N50     

Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators

Summary. The convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators, with focus on elliptic convection-diffusion operators in Sobolev space. Superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a Hilbert space. Then the same convergence result is verified for elliptic convection-diffusion equations using different preconditioning operators. The convergence factor involves the eigenvalues of the corresponding operators, for which an estimate is also given. The above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection-diffusion problems.Mathematics Subject Classification (2000): 65J10, 65F10, 65N15The second author was supported by the Hungarian Research Grant OTKA No. T. 043765.Dedicated to David M. Young on the occasion of his 80th birthday.     

A semi-discrete scheme for the stochastic nonlinear Schrödinger equation

Summary. We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies. Mathematics Subject Classification (2000):35Q55, 60H15, 65M06, 65M12     

Robust numerical methods for contingent claims under jump diffusion processes

An implicit method is developed for the numerical solution ofoption pricing models where it is assumed that the underlyingprocess is a jump diffusion. This method can be applied to avariety of contingent claim valuations, including American options,various kinds of exotic options, and models with uncertain volatilityor transaction costs. Proofs of timestepping stability and convergenceof a fixed-point iteration scheme are presented. For typicalmodel parameters, it is shown the error is reduced by two ordersof magnitude at each iteration. The correlation integral iscomputed using a fast Fourier transform method. Numerical testsof convergence for a variety of options are presented.     

Diagonal and Toeplitz splitting iteration methods for diagonal‐plus‐Toeplitz linear systems from spatial fractional diffusion equations

The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal‐plus‐Toeplitz structure. For solving these diagonal‐plus‐Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step‐size h is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant‐plus‐diagonal preconditioner proposed recently by Ng and Pan (M.K. Ng and J.‐Y. Pan, SIAM J. Sci. Comput. 2010;32:1442‐1464). Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show h‐independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big‐jump coefficients.     

Kinetic approximation of a boundary value problem for conservation laws

Summary. We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.Mathematics Subject Classification (2000): 65M06, 65M12, 76M20Correspondence to: D. Aregba-Driollet     

Quasi-likelihood analysis for the stochastic differential equation with jumps

In this paper, we consider a multidimensional diffusion process X with jumps whose jump term is driven by a compound Poisson process, and discuss its parametric estimation. We present asymptotic normality and convergence of moments of any order for a quasi-maximum likelihood estimator and a Bayes type estimator by assuming an exponential mixing property of X. To show these properties, we use the polynomial type large deviation theory.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

A posteriori error analysis for a class of integral equations and variational inequalities

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order ${2s \in (0,2]}We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ? (0,2]{2s \in (0,2]}. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H ^s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.     

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ^∞,2 (L ²) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions. Mathematics subject classifications (2000) 65M12, 65M25, 65M60, 92D25 M.-Y. Kim: This work was supported by Korea Research Foundation Grant (KRF-2001-041-D00037).     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

A priori error estimates for a mixed finite element discretization of the Richards’ equation

Summary. A mixed finite element discretization is applied to Richards equation, a nonlinear, possibly degenerate parabolic partial differential equation modeling water flow through porous medium. The equation is considered in its pressure formulation and includes both variably and fully saturated flow regime. Characteristic for such problems is the lack in regularity of the solution. To handle this we use a time-integrated scheme. We analyze the scheme and present error estimates showing its convergence.Mathematics Subject Classification (2000): 65M12, 65M60, 76S05, 35K65Acknowledgments. We would like to thank Markus Bause for very useful discussions and suggestions.     

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

Summary. We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation where generically for any given In addition to showing well-posedness of our approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented. Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 35K65, 35K35Supported by the EPSRC, U.K. through grant GR/M29689.Supported by the EPSRC, and by the DAAD through a Doktorandenstipendium     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

A second order accurate difference scheme for the heat equation with concentrated capacity

Summary. A numerical solution to the one-dimensional heat equation with concentrated capacity is considered. A second-order accurate difference scheme is derived by the method of reduction of order on non-uniform meshes. The solvability, stability and second order L convergence are proved. A numerical example demonstrates the theoretical results.Mathematics Subject Classification (2000): Primary 65M06, 65M12, 65M15The contract grant sponsor: National Natural Science Foundation of CHINA; The contract grant number:19801007     

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C ¹-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.