Time-dependent Partial Differential Equations Research Articles

RBFs Meshless Method of Lines for the Numerical Solution of Time-Dependent Nonlinear Coupled Partial Differential Equations

In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.

A General Framework for Deriving Integral Preserving Numerical Methods for PDEs

A general procedure for constructing conservative numerical integrators for time-dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities. The framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order. It is proved formally that second order convergence is obtained. The procedure is applied to a test case, and numerical experiments are provided.

MathPDE: A Package to Solve PDEs by Finite Differences

A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. It implements finite-difference methods. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability.

Domain decomposition operator splitting for mimetic finite difference discretizations of non-stationary problems

In this paper, we propose reliable and efficient numerical methods for solving semilinear, time-dependent partial differential equations of reaction–diffusion type. The original problem is first integrated in time by using a linearly implicit fractional step Runge–Kutta method. This method takes advantage of a suitable partitioning of the diffusion operator based on domain decomposition techniques. The resulting semidiscrete problem is fully discretized by means of a mimetic finite difference method on quadrilateral meshes. Due to the previous splitting, the totally discrete scheme can be reduced to a set of uncoupled linear systems which can be solved in parallel. The overall algorithm is unconditionally stable and second-order convergent in both time and space. These properties are confirmed by numerical experiments.

Preface

Preface

A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems

A code based on the two-stage Gauss formula (order four) for second-order initial value problems of a special type is developed. This code can be used to obtain a low- to medium-precision integration for a wide range of problems in the class of oscillatory type, Hamiltonian problems, and time-dependent partial differential equations discretized in space by finite differences or finite elements. The iteration process used in solving for the stage values of the Gauss formula, the selection of the initial step size, and the choice of an appropriate local error estimator for determining the step size change according to a particular tolerance specified by the user are studied. Moreover, a global error estimate and a dense output at equidistant points in the integration interval are supplied with the code. Numerical experiments and some comparisons with certain standard codes on relevant test problems are also given.

The Laplace transform method for Burgers' equation

AbstractThe Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, tk) and w(x, t), tk⩽t⩽tk+1, and obtain a time‐dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive. Copyright © 2009 John Wiley & Sons, Ltd.

A PDE pricing framework for cross-currency interest rate derivatives

Abstract We propose a general framework for efficient pricing via a Partial Differential Equation (PDE) approach of crosscurrency interest rate derivatives under the Hull–White model. In particular, we focus on pricing long-dated foreign exchange (FX) interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We formulate the problem in terms of three correlated processes that incorporate FX skew via a local volatility function. This formulation results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE. The Crank–Nicolson (CN) method and the Alternating Direction Implicit (ADI) method are considered for the time discretization. In the former case, the preconditioned Generalized Minimal Residual (GMRES) method is employed for the solution of the resulting block banded linear system at each time step, with the preconditioner solved by Fast Fourier Transform (FFT) techniques. Numerical results indicate that the numerical methods considered are second-order convergent, and, asymptotically, as the discretization granularity increases, almost optimal, with the ADI method being modestly more efficient than CNGMRES-FFT. An analysis of the impact of the FX volatility skew on the PRDC swaps’ prices is presented, showing that the FX volatility skew results in lower prices (i.e. profits) for the payer of PRDC coupons.

Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method

We investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM) in solving time dependent partial differential equations. To this effect we consider the homogeneous, non-homogeneous, linear and nonlinear Klein–Gordon equations with boundary conditions. The results reveal that the method is explicit, effective, and easy to use.

A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance

We study a parallel implementation on a Graphics Processing Unit (GPU) of Alternating Direction Implicit (ADI) time-discretization methods for solving time-dependent parabolic Partial Differential Equations (PDEs) in three spatial dimensions with mixed spatial derivatives in a variety of applications in computational finance. Finite differences on uniform grids are used for the spatial discretization of the PDEs. As examples, we apply the GPU-based parallel methods to price European rainbow and European basket options, each written on three assets. Numerical results showing the efficiency of the parallel methods are provided.

Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

Online parameter identification without Ricatti-type equations in a class of time-dependent partial differential equations: an extended state approach with potential to partial observations

Online parameter identification for dynamical systems is the task of estimating model parameters simultaneously to the time evolution of the physical state and plays a crucial role in adaptive control. The existing techniques for time-dependent partial differential equations exclude the case of partial state observations or require to solve online Ricatti-type operator evolution equations accompanied with high computational costs. In this paper we suggest and analyze an extended state approach to online parameter identification in a class of possibly nonlinear PDEs that avoids Ricatti-type equations but still has the potential to allow for partial state observations. Numerical examples underline the feasibility of our method.

GPU Pricing of Exotic Cross-Currency Interest Rate Derivatives with a Foreign Exchange Volatility Skew Model

We present a Graphics Processing Unit (GPU) parallelization of the computation of the price of exotic cross-currency interest rate derivatives via a Partial Differential Equation (PDE) approach. In particular, we focus on the GPU-based parallel pricing of long-dated foreign exchange (FX) interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with FX volatility skew which results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the Alternating Direction Implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture (CUDA) framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI timestepping technique. Numerical results indicate that GPUs can provide significant increase in performance over CPUs when pricing PRDC swaps. An analysis of the impact of the FX skew on such derivatives is provided.

A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations

Over the past years, model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models. A family of a priori model reduction techniques, called Proper Generalized Decomposition (PGD) methods, are receiving a growing interest. These methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces. They can be interpreted as generalizations of Proper Orthogonal Decomposition (POD) for the a priori construction of such separated representations. In this paper, we introduce and study different definitions of PGD for the solution of time-dependent partial differential equations. We review classical definitions of PGD based on Galerkin or Minimal Residual formulations and we propose and discuss several improvements for these classical definitions. We give an interpretation of optimal decompositions as the solution of pseudo-eigenproblems. We also introduce a new definition of PGD, called Minimax PGD, which can be interpreted as a Petrov–Galerkin model reduction technique, where test and trial reduced basis functions are related by an adjoint problem. This new definition improves convergence properties of separated representations with respect to a chosen metric. It coincides with a classical POD for degenerate time-dependent partial differential equations. For the numerical construction of each PGD, we propose algorithms inspired from the solution of eigenproblems. Several numerical examples illustrate and compare the different definitions of PGD on transient advection–diffusion–reaction equations.

An Efficient Numerical Method for the Solution of the $L_2$ Optimal Mass Transfer Problem

In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L(2) mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61-97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data.

Nonlinear Model Reduction via Discrete Empirical Interpolation

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh–Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives

We propose a general framework for efficient pricing via a Partial Differential Equation (PDE) approach of cross-currency interest rate derivatives under the Hull-White model. In particular, we focus on pricing long-dated foreign exchange (FX) interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We formulate the problem in terms of three correlated processes that incorporate FX skew via a local volatility function. This formulation results in a time dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE. The Crank-Nicolson (CN) method and the Alternating Direction Implicit (ADI) method are considered for the time discretization. In the former case, the preconditioned Generalized Minimal Residual (GMRES) method is employed for the solution of the resulting block banded linear system at each time step, with the preconditioner solved by Fast Fourier Transform (FFT) techniques. Numerical results indicate that the numerical methods considered are second-order convergent, and, asymptotically, as the discretization granularity increases, almost optimal, with the ADI method being modestly more efficient than CN-GMRES-FFT. An analysis of the impact of the FX volatility skew on the PRDC swaps' prices is presented, showing that the FX volatility skew results in lower prices (i.e. profits) for the payer of PRDC coupons.

Pricing of Cross-Currency Interest Rate Derivatives on Graphics Processing Units

We present a Graphics Processing Unit (GPU) parallelization of the computation of the price of cross-currency interest rate derivatives via a Partial Differential Equation (PDE) approach. In particular, we focus on the GPU-based parallel computation of the price of long-dated foreign exchange interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with foreign exchange skew which results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the Alternating Direction Implicit technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture (CUDA) framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU. Using this approach on two NVIDIA Tesla C870 GPUs of an NVIDIA 4-GPU Tesla S870 to price a Bermudan cancelable PRDC swap having a 30 year maturity and annual exchange of fund flows, we have achieved an asymptotic speedup by a factor of 44 relative to a single thread on a 2.0GHz Xeon processor.

On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion

Recently, a nonlocal term has been introduced in time-dependent partial differential equation (PDE) models of cell migration in tissue. This term is used to model adhesive effects between cells and also between cells and the extracellular matrix. We assume periodic boundary conditions for the model and that the PDE system is discretized following the method of lines and using a finite-volume scheme on a uniform grid in space. For high-resolution simulations of the PDE system an efficient evaluation of the approximation of the nonlocal term is crucial. For one and two spatial dimensions we develop suitable approximations of the nonlocal term and evaluate these using fast Fourier transform (FFT) techniques. Comprehensive numerical tests show the accuracy and efficiency of our approach. We also demonstrate the impact of the proposed scheme for the treatment of the nonlocal term on simulation times for a differential cell adhesion model. We discuss extensions and applicability of our work to systems with nonperiodic boundary conditions and for other nonlocal PDE models from mathematical biology.

How to adaptively resolve evolutionary singularities in differential equations with symmetry

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.