Time-dependent Partial Differential Equations Research Articles

Improved numerical solution of multi-asset option pricing problem: A localized RBF-FD approach

The objective of this work is to present a novel procedure for tackling European multi-asset option problems, which are modeled mathematically in terms of time-dependent parabolic partial differential equations with variable coefficients. To use as low as possible of number computational grid points, a non-uniform grid is generated while a radial basis function-finite difference scheme with the Gaussian function is applied on such a grid to discretize the model as efficiently as possible. To reduce the burdensome for tackling the resulting set of ordinary differential equations, a Krylov method, which is due to the application of exponential matrix function on a vector, is taken into account. The combination of these techniques reduces the computational effort and the elapsed time. Several experiments are brought froward to illustrate the superiority of the new improved approach. In fact, the contributed procedure is capable to tackle even 6D PDEs on a normally-equipped computer quickly and efficiently.

A novel space–time meshless method for nonhomogeneous convection–diffusion equations with variable coefficients

A new space–time meshless method is proposed for solving the nonhomogeneous convection–diffusion equations with variable coefficients. From the perspective of the space–time distance, the space–time radial basis function in a space and time scale framework is presented and then is employed to discretize the time-dependent parabolic partial differential equations. The present meshless scheme is a truly meshless method which requires neither domain or boundary discretization, and can be easily used for 1D, 2D, and 3D problems. Furthermore, the proposed new methodology does not require the discretization of the temporal derivatives, and therefore is very simple mathematically, relatively fast computationally, and easy to program. Numerical examples conform the effectiveness and accuracy of the proposed method.

Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

Determination of the thermal properties of a material is an important task in many scientific and engineering applications. How a material behaves when subjected to high or fluctuating temperatures can be critical to the safety and longevity of a system’s essential components. The laser flash experiment is a well-established technique for indirectly measuring the thermal diffusivity, and hence the thermal conductivity, of a material. In previous works, optimization schemes have been used to find estimates of the thermal conductivity and other quantities of interest which best fit a given model to experimental data. Adopting a Bayesian approach allows for prior beliefs about uncertain model inputs to be conditioned on experimental data to determine a posterior distribution, but probing this distribution using sampling techniques such as Markov chain Monte Carlo methods can be incredibly computationally intensive. This difficulty is especially true for forward models consisting of time-dependent partial differential equations. We pose the problem of determining the thermal conductivity of a material via the laser flash experiment as a Bayesian inverse problem in which the laser intensity is also treated as uncertain. We introduce a parametric surrogate model that takes the form of a stochastic Galerkin finite element approximation, also known as a generalized polynomial chaos expansion, and show how it can be used to sample efficiently from the approximate posterior distribution. This approach gives access not only to the sought-after estimate of the thermal conductivity but also important information about its relationship to the laser intensity, and information for uncertainty quantification. Moreover, this approach leads to significant speed up over traditional methods by orders of magnitude. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity.

Two-level implicit high order method based on half-step discretization for 1D unsteady biharmonic problems of first kind

In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients

For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework. For linear problems with constant coefficients, it is well-known in the literature that the choice of the numerical flux (e.g. central or upwind) and the selection of the polynomial basis (e.g. Gauß–Legendre or Gauß–Lobatto–Legendre) affects both the growth rate and the asymptotic value of the error. Here, we extend these investigations of the long time error to variable coefficients using both Gauß–Lobatto–Legendre and Gauß–Legendre nodes as well as several numerical fluxes. We derive conditions guaranteeing that the errors are still bounded in time. Furthermore, we analyse the error behaviour under these conditions and demonstrate in several numerical tests similarities to the case of constant coefficients. However, if these conditions are violated, the error shows a completely different behaviour. Indeed, by applying central numerical fluxes, the error increases without upper bound while upwind numerical fluxes can still result in uniformly bounded numerical errors. An explanation for this phenomenon is given, confirming our analytical investigations.

A numerical investigation of time-dependent MHD axisymmetric transport of Sisko fluid towards elongating porous disk

In this paper, we examined the unsteady boundary layer flow and heat transfer of a Sisko fluid model over an axisymmetric stretching porous disk in the presence of uniform magnetic field. Mathematical modelling is performed in cylindrical polar coordinates. By means of suitable transformations, the governing time-dependent partial differential equations are reduced to nonlinear coupled ordinary differential equations. Shooting method with Runge–Kutta of order 5 is employed to compute non-dimensional velocity and temperature. The effects of pertinent parameters are portrayed through graphs. The skin friction coefficient and Nusselt number are tabulated to study the behaviours at the stretching surface.

Numerical solution through mathematical modelling of unsteady MHD flow past a semi-infinite vertical moving plate with chemical reaction and radiation

Abstract In the present manuscript, unsteady magnetohydrodynamic (MHD) flow over a moving porous semi-infinite vertical plate with time-dependent suction has been studied in the presence of chemical reaction and radiation parameters. Time-dependent partial differential equations in the dimensionless form are solved numerically through mathematical modelling in COMSOL Multiphysics. The results are obtained for velocity, temperature and concentration profiles at different times. Steady state results are also presented for different values of physical parameters. The parameters involved in the problem are useful to change the characteristics of velocity, heat transfer and concentration profiles. The numerical solution of partial differential equations involved in the problem is obtained without sacrificing the relevant physical phenomena.

Pricing the financial Heston–Hull–White model with arbitrary correlation factors via an adaptive FDM

This paper is concerned with the pricing procedure of one of the most challenging models known as the Heston–Hull–White partial differential equation (PDE) in option pricing, at which the model is a time-dependent 3D linear PDE including three mixed derivative terms. The model comes from the fact that the price, the volatility and the interest rate are assumed to be stochastic processes. To contribute and avoid huge discretized systems, an adaptive distribution of the nodes (viz, nonuniform nodes) is taken into account with emphasis on the hot area of the solution curve. New adaptive finite difference (FD) formulas of higher orders are constructed on these meshes. Then, a set of semi-discretized equations is attained which is tackled by a time-stepping method. Several financial tests are discussed in detail to reveal the superiority of the proposed approach.

A numerical treatment of unsteady three-dimensional hydromagnetic flow of a Casson fluid with Hall and radiation effects

Abstract The study of the unsteady magnetohydrodynamic three-dimensional radiative flow of a viscous incompressible electrically conducting Casson liquid along a stretching surface including the effects of Hall current is presented in this article. The fluid flow model consists of time-dependent partial differential equations which are highly non-linear. Similarity transformations are utilized to obtain ordinary differential equations in similarity form. Further, a numerical approach, namely spectral quasilinearization method (SQLM), is used to solve resulting highly nonlinear ordinary differential equations. A detailed analysis is carried out to study the influences of significant parameters, such as Casson liquid parameter, Hall current parameter, magnetic parameter, unsteadiness parameter, radiative parameter, on the profiles’ of the velocity field and temperature field. The behavior of emerging quantities of engineering interest such as skin friction coefficient and the Nusselt number is also studied. Fluid flow model as presented in the paper finds applications in silicon suspensions, blood flow, polymer engineering, and printing industry.

Numerical Study on an Unsteady Flow of an Immiscible Micropolar Fluid Sandwiched Between Newtonian Fluids Through a Channel

This paper deals with an unsteady flow of a micropolar fluid sandwiched between Newtonian fluids through a horizontal channel. The governing time-dependent partial differential equations are solved numerically by using the Crank-Nicolson finite difference approach. The continuity of velocity and shear stress is considered at the fluid–fluid interfaces. It is observed that the fluid velocities increase with time; eventually, a steady state is reached at a certain time instant. The velocity decreases with increasing micropolarity parameter in the micropolar fluid region and remains almost unchanged in both Newtonian fluid regions.

A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability.

Modeling spatially dependent functional data via regression with differential regularization

We propose a method for modeling spatially dependent functional data, based on regression with differential regularization. The regularizing term enables to include problem-specific information about the spatio-temporal variation of the phenomenon under study, formalized in terms of a time-dependent partial differential equation. The method is implemented using a discretization based on finite elements in space and finite differences in time. This non-tensor product basis allows to handle efficiently data distributed over complex domains and where the shape of the domain influences the phenomenon’s behavior. Moreover, the method can comply with specific conditions at the boundary of the domain of interest. Simulation studies compare the proposed model to available techniques for spatio-temporal data. The method is also illustrated via an application to the study of blood-flow velocity field in a carotid artery affected by atherosclerosis, starting from echo-color doppler and magnetic resonance imaging data.

Adaptive moving knots meshless method for simulating time dependent partial differential equations

This paper presents an improvement of Huang’s moving knots method (Huang et al.) for solving the PDE on the contours of Equidistribution Principle (EP-contours). Firstly, a new moving knots strategy is generated by discovering a significant factor ignored (dropped) by Huang (Huang et al.). The proposed strategy ensures that arbitrary initial knots could asymptotically converge to the EP-contours. Then the moving knots strategy and PDE are simulated by using multi-quadric (MQ) quasi-interpolation step by Step (3.3, 3.4). Error estimates of the algorithm are given. At last, numerical experiments are provided to illustrate the validity of the algorithm. Both theoretical analysis and numerical results show that the proposed algorithm benefits the methods in Huang et al. and Wu.

Conservative or dissipative quasi-interpolation method for evolutionary partial differential equations

Based on quasi-interpolation, we propose a new meshless conservative or dissipative method for nonlinear time-dependent partial differential equations. Using the method of lines, we first discretize the equation in space with the quasi-interpolation method, then employ the average vector field method in time discretization to derive the final numerical scheme. The method not only inherits the conservation or dissipation property of the equation but also has the meshless feature since we use the nonuniform grids in spatial discretization. Several numerical examples are presented to demonstrate the effectiveness of the proposed method.

Diagonalization of 1-D differential operators with piecewise constant coefficients using the uncertainty principle

A highly accurate and efficient numerical method is presented for computing the solution of a 1-D time-dependent partial differential equation in which the spatial differential operator features a piecewise constant coefficient defined on n pieces, in either self-adjoint and non-self-adjoint form, on a finite interval with periodic boundary conditions. The Uncertainty Principle is used to estimate the eigenvalues of the operator. Then, these estimates are used to construct a basis of eigenfunctions for use with a spectral method. The solution is presented as a truncated eigenfunction expansion, where each eigenfunction is a wave function that changes frequencies at the interfaces between different materials. Numerical experiments demonstrate the accuracy, efficiency and scalability of the method in comparison to other methods.

A Moving Mesh Method for Mathematical Model of Capillary Formation in Tumor Angiogenesis

Using adaptive mesh methods is one of the strategies to improve numerical solutions in time-dependent partial differential equations. Moving mesh method is an adaptive mesh method. In this method, an increase in the number of mesh points is not needed for increasing the accuracy and efficiency of numerical solutions. Highly accurate solutions only can be obtained by concentrating points in areas where more accuracy is needed and decreasing the number of points in smooth areas. In this paper, moving mesh method is applied for the mathematical model of capillary formation in tumor angiogenesis. Numerical results show that this method leads to stable solutions for large values of cell diffusion coefficient in addition to enjoying necessary accuracy and efficiency.

DYNAMICAL BEHAVIOR OF A HEAT PUMP COAXIAL EVAPORATOR CONSIDERING THE PHASE BORDER’S IMPACT ON CONVERGENCE

Using a dynamical mathematical model, we investigated transient behavior of a water-water heat pump’s evaporator. The model consists of time and space dependent partial differential equations of water, pipe wall and refrigerant. Mathematically the thermal expansion valve (TEV) and compressor are described with lumped parameters and represent the boundary conditions. During the numerical solution of this system of equations the problem emerged of divergence of solutions. It was determined that the cause of the divergence solution was in the location of phase change of the refrigerant. The aim of this paper is, firstly, to display and propose a new approach to the elimination of divergence. In addition, it examines dynamic behavior of the heat pumps’ coaxial evaporator.

Optimal sensor placement for joint parameter and state estimation problems in large-scale dynamical systems with applications to thermo-mechanics

We consider large-scale dynamical systems in which both the initial state and some parameters are unknown. These unknown quantities must be estimated from partial state observations over a time window. A data assimilation framework is applied for this purpose. Specifically, we focus on large-scale linear systems with multiplicative parameter-state coupling as they arise in the discretization of parametric linear time-dependent partial differential equations. Another feature of our work is the presence of a quantity of interest different from the unknown parameters, which is to be estimated based on the available data. In this setting, we employ a simplicial decomposition algorithm for an optimal sensor placement and set forth formulae for the efficient evaluation of all required quantities. As a guiding example, we consider a thermo-mechanical PDE system with the temperature constituting the system state and the induced displacement at a certain reference point as the quantity of interest.

Some Recent Developments in Superconvergence of Discontinuous Galerkin Methods for Time-Dependent Partial Differential Equations

In this paper, we briefly review some recent developments in the superconvergence of three types of discontinuous Galerkin (DG) methods for time-dependent partial differential equations: the standard DG method, the local discontinuous Galerkin method, and the direct discontinuous Galerkin method. A survey of our own results for various time-dependent partial differential equations is presented and the superconvergence phenomena of the aforementioned three types of DG solutions are studied for: (i) the function value and derivative approximation at some special points, (ii) cell average error and supercloseness.

GPU acceleration of time gating based reverse time migration using the pseudospectral time-domain algorithm

We present a Graphics Processing Units (GPU) implementation of time gating based reverse time migration (TG-RTM) which uses the pseudospectral time-domain (PSTD) algorithm to solve the acoustic wave equation. TG-RTM adopts the prior information of surrounding media to strengthen the correlation between the wavefields, thus has advantages in locating the targets over traditional reverse time migration (RTM) methods. The PSTD algorithm adopts fast Fourier transform (FFT) to obtain the spatial derivatives and a perfectly matched layer as an absorbing boundary condition to eliminate the wraparound effect introduced by the FFT periodicity assumption. Under the Nyquist sampling theorem, the spatial sampling density of the PSTD algorithm requires only two points per minimum wavelength. Thus, the PSTD algorithm can solve the time dependent partial differential equations efficiently and save mass computer memory. Compared with traditional RTM based on the finite difference time domain (FDTD) algorithm, the proposed RTM based on the PSTD algorithm can be implemented on a memory-limited GPU and can solve much larger models. To secure a better performance and generality of FFT in GPU, we present a scheme which combines 1D FFT with matrix transpositions instead of using 3D FFT directly. The matrix transpositions use shared memory to improve memory access efficiency. We also apply an efficient FFT scheme which replaces even-sized R2C FFT with a half-sized C2C FFT. For a small amount and balanced memory swapping from computer to GPU, we save the boundaries in lieu of checkpointing scheme when we propagate the source wavefield forward and backward. The proposed RTM has an acceleration ratio of about 80 times by a Tesla K20X GPU card on a desktop computer. The simulation results of 2D and 3D models demonstrate that the proposed RTM is fast and inexpensive.