Fully-implicit Method Research Articles

Coupling-strength criteria for sequential implicit formulations

The Fully Implicit Method (FIM) is used widely in the reservoir simulation community, especially for the so-called black-oil model. The primary reason for resorting to FIM is its unconditional stability, but this comes at the high computational cost of solving coupled nonlinear systems of all the conservation equations simultaneously. In addition, having a scalable FIM simulator relies on specialized multi-stage linear solvers. In the last few years, the Sequential Fully Implicit (SFI) formulation has received a great deal of attention. SFI decomposes the fully coupled nonlinear problem into two nonlinear sub-problems: flow and transport. The flow problem is solved implicitly for pressure, and the transport problem is solved implicitly for the saturation(s). The splitting usually assumes that the total-velocity is fixed during the transport computations. Because the pressure has a near-elliptic behavior and the saturation(s) have near-hyperbolic behavior, the flow-transport splitting of SFI allows for the use of dedicated scalable nonlinear and linear solvers for each sub-problem.At convergence, SFI and FIM lead to the same numerical solution within the specified tolerances on the residuals. SFI, however, can suffer from convergence difficulties of the overall nonlinear iterations between flow and transport. The convergence difficulties of SFI are intimately related to the splitting, and they depend on the type and strength of coupling between the nonlinear flow and transport problems. In this paper, we develop indicators that quantify the nature and strength of coupling across the systems of equations and solution variables. The indicators quantify the coupling in both space and time. Our strategy employs terms in the FIM Jacobian matrix to quantifying the sensitivity of the mass conservation equations and the global volume-balance to changes in pressure and saturation.These ‘coupling strength’ indicators are used in a domain-decomposition strategy to define local sub-domains for which a fully coupled correction is performed instead of the usual fixed-point iteration update. We demonstrate the accuracy and utility of these indicators for several challenging problems, including buoyancy driven lock-exchange - with or without gas solubility, as well as, gas and water injection problems in heterogeneous reservoir models.

Accelerated nonlinear domain decomposition solver for multi-phase flow and transport in porous media

Development of robust and efficient nonlinear solution strategies for multi-phase flow and transport within natural porous media is challenging. Fully Implicit Method (FIM) is widely used in reservoir simulation for solving nonlinear algebraic systems. Previous studies showed that nonlinearities of flow and transport problems could exhibit large degrees of locality across timesteps and nonlinear iterations. Nonlinear domain decomposition (NDD) is an attractive class of methods for achieving localization and resolving spatially unbalanced nonlinearities. However, NDD is not robust as iterative solvers, because the outer iterations are essentially fixed-point type.We design a solution framework to accelerate the outer-loop convergence of NDD. We first apply a nonlinear acceleration (NA) technique to the fixed-point iterations, for correcting the boundary conditions of subdomains. Because flow problems are parabolic and exhibit global behaviors, long-range information is essential to capture main features of the flow field. We incorporate a preconditioning step based on CPR-AMG before each timestep, to provide better initial solutions. Additionally, smoother iterations can be taken to improve the boundary conditions during the NDD process.We evaluate the NDD methods using heterogeneous multi-phase models. The nonlinearities associated with the flow, transport and the coupling are locally resolved through the NDD iterations. Results demonstrate that the basic NDD solver is afflicted with severe nonlinear difficulties. By comparison, the accelerated NDD solver greatly improves the outer convergence performance. Moreover, the new solver can effectively exploit locality on the Newton level, for both the flow and transport problems.

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach—and in particular, FSMSN—results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance–i.e., measured by wall-clock time–will be studied in a subsequent publication.

Efficient dissipation-based nonlinear solver for multiphase flow in discrete fractured media

Solving nonlinear equation systems resulting from the Fully Implicit Method (FIM) remains a challenge for numerical simulation of multiphase flow in subsurface fractured media. The Courant numbers can vary orders of magnitude across discrete fracture-matrix (DFM) models because of the high contrasts in the permeability and length-scale between matrix and fracture. The standard Newton solver is usually unable to converge for big timestep sizes.Limited research has been conducted on nonlinear solver techniques for multiphase flow and transport in fractured media. We make an extension of a new dissipation-based continuation (DBC) method to DFM models. Our goal is to prevent timestep cuts and maintain efficient time-stepping for FIM. The DBC algorithm builds a homotopy of the discretized conservation equations through the addition of numerical dissipation terms. We introduce a continuation parameter for controlling the dissipation and ensuring that accuracy of the computed solution will not be reduced. Under the nonlinear framework of DBC, general dissipation operators and adaptive strategies are developed to provide the optimal dissipation matrix for the multiphase hyperbolic system.We assess the new nonlinear solver using multiple numerical examples. Results reveal that the damped-Newton solver suffers from serious restrictions on timestep sizes and wasted iterations. In contrast, the DBC solver provides excellent computational performance. The dissipation operators are able to successfully resolve main convergence difficulties. We also investigate the impact of star-delta transformation which removes the small cells at fracture intersections. Moreover, we demonstrate that an aggressive time-stepping does not affect the solution accuracy.

Parallel fully implicit chemical potential-based modeling of unconventional shale gas reservoirs

The design of novel mathematical models and state-of-the-art simulators for unconventional shale gas reservoirs plays an increasingly important role in the current energy mix of the world’s growing energy demand. In this paper, we present a robust and scalable framework, which is based on the chemical potential-based modeling and the fully implicit method (FIM), to model and simulate this highly nonlinear flow transport problem on a large scale. In the proposed approach, a thermodynamically consistent mathematical model of gas flow in shale formations, which employs the gas density and the chemical potential gradient, is developed to satisfy the second law of thermodynamics and meanwhile guarantee the energy dissipation property. And then the family of fully implicit finite element algorithms is utilized to accurately capture the complicated flow physics behind the transport process in shale media on high resolution grids. In particular, our approach further enhances the numerical formulation by proposing the family of Newton–Krylov methods for efficiently computing, and the parallel implementation of the simulator is achieved by using the domain decomposition technique. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the solution strategies for several interesting shale gas flow problems in two or three dimensions. With the proposed parallel method, large-scale reservoir simulation can be obtained on the Shaheen-II supercomputer with up to 40960 processors, which enables to achieve a good strong scalability by saving more than 90 percent of computing time, when the problem size is enlarged to hundreds of millions of degrees of freedom.

A MODIFIED SEQUENTIAL FULLY IMPLICIT METHOD OF COMPOSITIONAL RESERVOIR SIMULATION FOR CO2 FLOODING IN ADAPTIVE MESH REFINEMENT GRIDS

A MODIFIED SEQUENTIAL FULLY IMPLICIT METHOD OF COMPOSITIONAL RESERVOIR SIMULATION FOR CO2 FLOODING IN ADAPTIVE MESH REFINEMENT GRIDS

ICE: Implicit Coordinate Encoder for Multiple Image Neural Representation.

In recent years, implicit neural representations (INR) have shown their great potential to solve many computer graphics and computer vision problems. With this technique, signals such as 2D images or 3D shapes can be fit by training multi-layer perceptrons (MLP) on continuous functions, providing many advantages over conventional discrete representations. Despite being considered a promising approach to 2D image encoding and compression, the application of INR to image collections remains a challenge, since the number of parameters needed rapidly grow with the number of images. In this paper, we propose a fully implicit approach to INR which drastically reduces the size of MLP models in multiple image representation tasks. We introduce the concept of implicit coordinate encoder (ICE) and show it can be used to scale INR with the image number; specifically, by learning a common feature space between images. Furthermore, we show that our method is valid not only for image collections but also for large (gigapixel) images by applying a "divide-and-conquer" strategy. We propose an auto-encoder deep neural network architecture, with a single ICE (encoder) and multiple MLP (decoders), which are jointly trained following a multi-task learning strategy. We demonstrate the benefits coming from ICE when it is implemented as a one-dimensional convolutional encoder, including a better performance of the downstream MLP models with an order of magnitude fewer parameters. Our method is the first one to make use of convolutional blocks in INR networks, unlike the conventional approach of using MLP architectures only. We show the benefits of ICE in two experimental scenarios: a collection of twenty-four small ( 768×512 ) images (Kodak dataset), and a single large ( 3072×3072 ) image (dwarf planet Pluto), achieving better quality than previous fully-implicit methods, using up to 50% fewer parameters.

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

Adaptive Timestepping, Linearization, and A Posteriori Error Control for Multiphase Flow of Immiscible Fluids in Porous Media with Wells

Summary This work focuses on the development of adaptive timesteps, stopping criteria, and error control strategies for reservoir simulations with fully implicit (FIM) solvers. Using a rigorous error control framework, an adaptive time selector combined with nonlinear stopping criteria is used to control nonlinear iterations as well as to balance accuracy and robustness for challenging nonlinear simulations. In reservoir simulation, efficiently solving a system of nonlinear equations arising from the FIM method can be computationally burdensome for complex recovery processes. Theoretically, an FIM reservoir simulator has no stability limit on the timestep size. In practice, standard Newton’s method often fails to converge for large timestep sizes and must therefore cut the timestep multiple times to achieve convergence, resulting in a large number of unnecessary iterations. Another cause of nonlinear convergence issues is the presence of wells, which are often presented as singular point/line sources strongly coupled to the reservoir model, posing additional restrictions on the timestep choice. Here, we use a posteriori error estimators to avoid unnecessary nonlinear iterations and timestep cuts when solving immiscible multiphase flow. First, we estimate error components (e.g., spatial, temporal, and nonlinear) and then apply these to balancing criteria, providing us with dynamic and adaptive strategies to control timestep and nonlinear iterations. The error estimators are fully and locally computable, inexpensive to use, and target the various error components, including well singularities. The method provides an adaptive criterion for stopping the nonlinear iteration process whenever the linearization error does not significantly affect the overall error. Simultaneously, timesteps are adapted to maintain a constant size of the temporal discretization error with respect to the total error. Altogether, this avoids using unnecessary linearization iterations, wasteful timestep cuts, and too small timesteps. To demonstrate the effectiveness of these adaptive features, we present results for a suite of cases, covering both standard benchmarks and conceptual problems incorporating highly heterogeneous media with multiple wells. The proposed timestep selector cooperates with the new stopping criteria to improve nonlinear solver performance and increases robustness for cases with high nonlinearity. Perhaps most important, the adaptive features ensure balanced temporal and spatial errors while maintaining sufficiently small nonlinear errors, which ensures solution accuracy by accurately reproducing saturation fronts, production plateau, and breakthrough times.

Dissipation-Based Nonlinear Solver for Fully Implicit Compositional Simulation

Summary In reservoir simulation of complex recovery processes, solving nonlinear equation system arising from the fully implicit method (FIM) poses a significant challenge. The Courant-Friedrichs-Lewy (CFL) conditions can span orders of magnitude in heterogeneous reservoir models because of large variations of permeability and porosity. The standard Newton method often fails to converge for large timestep sizes. We extend a new dissipation-based continuation (DBC) method to multiphase and compositional problems. The objective is to avoid timestep cuts and maintain efficient timestepping in FIM simulations. The method constructs a homotopy of the discrete governing equations by adding numerical dissipation. A continuation parameter is introduced to control the dissipation level and ensure that the accuracy of the converged solution is not degraded. Within the DBC framework, we develop general dissipation operators for multiphase and compositional flow models. Adaptive strategies are also proposed to determine the optimum dissipation matrix for hyperbolic systems. We evaluate the efficiency of the new nonlinear solver using several challenging cases. Results show that the standard damped Newton solver is afflicted by severe timestep restrictions and wasted computations. By comparison, the DBC solver enables superior nonlinear convergence. The dissipation operators can effectively overcome major convergence difficulties of coupled flow and transport problems.

A Graphics Processing Unit–Based, Industrial Grade Compositional Reservoir Simulator

SummaryRecently, graphics processing units (GPUs) have been demonstrated to provide a significant performance benefit for black-oil reservoir simulation, as well as flash calculations that serve an important role in compositional simulation. A comprehensive approach to compositional simulation based on GPUs has yet to emerge, and the question remains as to whether the benefits observed in black-oil simulation persist with a more complex fluid description. We present a positive answer to this question through the extension of a commercial GPU-based black-oil simulator to include a compositional description based on standard cubic equations of state (EOSs). We describe the motivations for the selected nonlinear formulation, including the choice of primary variables and iteration scheme, and support for both fully implicit methods (FIMs) and adaptive implicit methods (AIMs). We then present performance results on an example sector model and simplified synthetic case designed to allow a detailed examination of runtime and memory scaling with respect to the number of hydrocarbon components and model size, as well as the number of processors. We finally show results from two complex asset models (synthetic and real) and examine performance scaling with respect to GPU generation, demonstrating that performance correlates strongly with GPU memory bandwidth.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

An Adaptive Sequential Fully Implicit Domain-Decomposition Solver

SummaryModern reservoir simulation must handle complex compositional fluid behavior, orders-of-magnitude variations in rock properties, and large velocity contrasts. We investigate how one can use nonlinear domain-decomposition preconditioning to combine sequential and fully implicit (FI) solution strategies to devise robust and highly efficient nonlinear solvers. A full simulation model can be split into smaller subdomains that each can be solved independently, treating variables in all other subdomains as fixed. In subdomains with weaker coupling between flow and transport, we use a sequential fully implicit (SFI) solution strategy, whereas regions with stronger coupling are solved with an FI method. Convergence to the FI solution is ensured by a global update that efficiently resolves long-range interactions across subdomains. The result is a solution strategy that combines the efficiency of SFI and its ability to use specialized solvers for flow and transport with the robustness and correctness of FI. We demonstrate the efficacy of the proposed method through a range of test cases, including both contrived setups to test nonlinear solver performance and realistic field models with complex geology and fluid physics. For each case, we compare the results with those obtained using standard FI and SFI solvers.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

Sequential fully implicit Newton method for compositional flow and transport

Compositional simulation for flow and transport in porous media is essential for modeling Enhanced Oil Recovery (EOR) processes. The Sequential Fully Implicit (SFI) scheme is considered as a more flexible and potentially more computationally efficient alternative to the commonly used Fully Implicit Method (FIM). SFI was proposed along with the multiscale method where the flow and transport problems are solved in sequence. SFI opens the door for flexible discretizations, specialized linear solvers, and adaptive strategies. In the SFI method, a time step entails several ‘outer’ iterations. Each iteration consists of two steps: (1) a pressure equation is formed and solved implicitly, and then (2) a set of transport equations is solved, also implicitly. The (outer) iterations are conducted in a fixed-point manner until global convergence is reached. SFI can suffer from slow convergence, or even convergence failure, in strongly coupled flow and transport problems. Examples include dominant buoyancy and complex phase behaviors which are typical in compositional processes. Wong et al. [40] proposed the Sequential Fully Implicit Newton (SFIN) method for multi-physics problems in reservoir simulation and achieved superior outer loop convergence. In this work, we develop the SFIN method for compositional flow and transport problems, and in particular, show how the main assumption of fixing the total velocity can be incorporated. The key step of the SFIN method is a global Newton update that is performed at the end of each flow-transport iteration. During this outer Newton step, the pressure and transport variables are updated together to resolve the coupling between them better. The corresponding linear system is solved by a Krylov solver (e.g., GMRES) using a Jacobian-free approach that only entails matrix-vector multiplications. The latter is constructed by reusing previously computed Jacobian matrices and their preconditioners' information in a computationally efficient way. We present several challenging two- and three-dimensional examples and show that the SFIN method can significantly reduce the number of outer and inner loop iterations compared with the original (non-accelerated) SFI method. We analyze both the nonlinear and linear convergence behaviors for the SFIN method's key components and provide potential speedup estimates with respect to the SFI method.

Stabilized material point methods for coupled large deformation and fluid flow in porous materials

The material point method (MPM) has been increasingly used for the simulation of large-deformation processes in fluid-infiltrated porous materials. For undrained poromechanical problems, however, standard MPMs are numerically unstable because they use low-order interpolation functions that violate the inf–sup stability condition. In this work, we develop stabilized MPM formulations for dynamic and quasi-static poromechanics that permit the use of standard low-order interpolation functions notwithstanding the drainage condition. For the stabilization of both dynamic and quasi-static formulations, we utilize the polynomial pressure projection method whereby a stabilization term is augmented to the balance of mass. The stabilization term can be implemented with both the original and generalized interpolation material point (GIMP) methods, and it is compatible with existing time-integration methods. Here we use fully-implicit methods for both dynamic and quasi-static poromechanical problems, aided by a block-preconditioned Newton–Krylov solver. The stabilized MPMs are verified and investigated through several numerical examples under dynamic and quasi-static conditions. Results show that the proposed MPM formulations allow standard low-order interpolation functions to be used for both the solid displacement and pore pressure fields of poromechanical formulations, from undrained to drained conditions, and from dynamic to quasi-static conditions.

Unified thermo-compositional-mechanical framework for reservoir simulation

We present a reservoir simulation framework for coupled thermal-compositional-mechanics processes. We use finite-volume methods to discretize the mass and energy conservation equations and finite-element methods for the mechanics problem. We use the first-order backward Euler for time. We solve the resulting set of nonlinear algebraic equations using fully implicit (FI) and sequential-implicit (SI) solution schemes. The FI approach is attractive for general-purpose simulation due to its unconditional stability. However, the FI method requires the development of a complex thermo-compositional-mechanics framework for the nonlinear problems of interest, and that includes the construction of the full Jacobian matrix for the coupled multi-physics discrete system of equations. On the other hand, SI-based solution schemes allow for relatively fast development because different simulation modules can be coupled more easily. The challenge with SI schemes is that the nonlinear convergence rate depends strongly on the coupling strength across the physical mechanisms and on the details of the sequential updating strategy across the different physics modules. The flexible automatic differentiation-based framework described here allows for detailed assessment of the robustness and computational efficiency of different coupling schemes for a wide range of multi-physics subsurface problems.

On the efficiency of the IMPES method for two phase flow problems in porous media

In this work, we show a method to efficiently solve the multiphase flow through a porous media in the near wellbore region. The numerical discretisation is based on the IMplicit Pressure Explicit Saturation (IMPES) approach.While most of the works in the literature rely on Fully Implicit Methods (FIM) to simulate the reservoir, this is not suited for the near wellbore region, where smaller computational and physical times are required, therefore parametric models with history matching are used in this zone. However, parametric models present several uncertainties that can affect the estimation of the pressure drop. An accurate and fast simulation of that region is therefore required.Here, we propose a new and robust implementation of the IMPES scheme, aiming to reduce the large computational cost of the (implicit) pressure system of equations and increase the robustness and reliability of the scheme. The method takes advantage of the short physical time steps between iterations, observed in the near wellbore region, to produce a new pressure field at the cost of a numerical explicit scheme. Two and three dimensional numerical experiments are carried out to show the validity of the proposed discretisation. We show that the cost of the method is reduced in a factor of 7–19 when compared to the classical IMPES.

Modified sequential fully implicit scheme for compositional flow simulation

The fully implicit (FI) method is widely used for numerical modeling of multiphase flow and transport in porous media. The FI method is unconditionally stable, but that comes at the cost of a low-order approximation and high computational cost. The FI method entails iterative linearization and solution of fully-coupled linear systems with mixed elliptic/hyperbolic character. However, in methods that treat the near-elliptic (flow) and hyperbolic (transport) separately, such as multiscale formulations, sequential solution strategies are used to couple the flow (pressures and velocities) and the transport (saturations/compositions). The most common sequential schemes are: the implicit pressure explicit saturation (IMPES), and the sequential fully implicit (SFI) schemes. Problems of practical interest often involve tightly coupled nonlinear interactions between the multiphase flow and the multi-component transport. For such problems, the IMPES approach usually suffers from prohibitively small timesteps in order to obtain stable numerical solutions. The SFI method, on the other hand, does not suffer from a temporal stability limit, but the convergence rate can be extremely slow. This slow convergence rate of SFI can offset the gains obtained from separate and specialized treatments of the flow and transport problems. In this paper, we analyze the nonlinear coupling between flow and transport for compressible, compositional systems with complex interphase mass transfer. We isolate the nonlinear effects related to transmissibility and compressibility from those due to interphase mass transfer, and we propose a modified SFI (m-SFI) method. The new scheme involves enriching the ‘standard’ pressure equation with coupling between the pressure and the saturations/compositions. The modification resolves the convergence problems associated with SFI and provides a strong basis for using sequential formulations for general-purpose simulation. For a wide parameter range, we show that the proposed m-SFI algorithm is as robust as the FI method with a comparable convergence rate.

Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow

The Fully Implicit Method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media. The FIM involves solving large coupled systems of nonlinear algebraic equations. Newton-based methods, which are employed to solve the nonlinear systems, can suffer from convergence problems—this is especially true for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the time step is usually reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok and Tchelepi, J. Comput. Phys. 227(1), 706–727 9). Here, we improve the robustness of the potential-based ordering method in the presence of gravity. Furthermore, we also extend this nonlinear approach to interphase mass transfer. Our algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas coming out of solution and forming a separate phase), as a function of pressure and composition. Detailed comparisons of the robustness and efficiency of the potential-based solver with state-of-the-art nonlinear/linear solvers are presented for immiscible two-phase (Dead-Oil), Black-Oil, and compositional problems using heterogeneous models. The results show that for large time steps, our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which leads to gains in the overall computational cost.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

A new finite difference method for pricing and hedging fixed income derivatives: Comparative analysis and the case of an Asian option

We propose a second order accurate numerical finite difference method to replace the classical schemes used to solve PDEs in financial engineering. We name it Modified Fully Implicit method. The motivation for doing so stems from the accuracy loss while trying to stabilize the solution via the up-wind scheme in the convective term as well as the fact that spurious oscillations solutions occur when volatilities are low (this is actually the range that is commonly observed in interest rate markets). Unlike the classical schemes, our method covers the whole spectrum of volatilities in the interest rate dynamics.We obtain analytical and numerical results for pricing and hedging a zero-coupon bond and an Asian interest rate option. In the case of the Asian option, we compare the realistic discrete compounding interest rate scheme (associated with the Modified Fully Implicit method) with the continuous compounding scheme (often exploited in the literature), obtaining relative discrepancies between prices exceeding 50%. This indicates that the former scheme is more appropriate then the latter to price more complicate derivatives than straight bonds.