Waveform Relaxation Technique Research Articles

One‐Step Family of Three Optimized Second‐Derivative Hybrid Block Methods for Solving First‐Order Stiff Problems

This paper introduces a novel approach for solving first‐order stiff initial value problems through the development of a one‐step family of three optimized second‐derivative hybrid block methods. The optimization process was integrated into the derivation of the methods to achieve maximal accuracy. Through a rigorous analysis, it was determined that the methods exhibit properties of consistency, zero‐stability, convergence, and A‐stability. The proposed methods were implemented using the waveform relaxation technique, and the computed results demonstrated the superiority of these schemes over certain existing methods investigated in the study.

Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation

This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes. Non-uniform grids are essential to capture the singularities of “typical” solutions of time-fractional problems. We propose an efficient space-time multigrid method based on the waveform relaxation technique, which accounts for the nonlocal character of the fractional differential operator. To maintain an optimal complexity, which can be obtained for the case of uniform grids, we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix (H-matrix) representation. In particular, the proposed method has a computational cost of O(kNMlog⁡(M)), where M is the number of time steps, N is the number of spatial grid points, and k is a parameter which controls the accuracy of the H-matrix approximation. The efficiency and the good convergence of the algorithm, which can be theoretically justified by a semi-algebraic mode analysis, are demonstrated through numerical experiments in both one- and two-dimensional spaces.

Efficient Simulation of Field/Circuit Coupled Systems With Parallelized Waveform Relaxation

This paper proposes an efficient parallelised computation of field/circuit coupled systems co-simulated with the Waveform Relaxation (WR) technique. The main idea of the introduced approach lies in application of the parallel-in-time method parareal to the WR framework. Acceleration obtained by the time-parallelisation is further increased in the context of micro-macro parareal. Here, the field system is replaced by a lumped model in the circuit environment for the sequential computations of parareal. The introduced algorithm is tested with a model of a single-phase isolation transformer coupled to a rectifier circuit.

A time-domain method for the electromagnetic transient response of multiconductor transmission lines excited by the electromagnetic pulse

An efficient field-to-line coupling method in the time-domain is developed based on the waveform relaxation (WR) techniques for evaluating the electromagnetic(EM) transient response of multiconductor transmission lines (MTLs) excited by an electromagnetic pulse(EMP). Firstly, the incident EMP electric fields on the lines are calculated as equivalent distributed sources introduced into the MTLs equations. Then, the telegraphs equations of lossless transmission lines are decoupled by applying the WR techniques and different types of equivalent current sources at the terminal of MTLs are calculated using the iterative convolution and analytical method separately. Finally, the equivalent circuit is established and the transient currents/voltages induced on the lines are obtained by using the circuit simulation software PSCAD. The proposed time-domain method has the advantages of accuracy and high efficiency compared with the method based on finite-difference time-domain (FDTD) via two numerical examples.

Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique

We present in this paper a time parallel algorithm for $${\dot{u}}=f(t,u)$$ with initial-value $$u(0)=u_0$$ , by using the waveform relaxation (WR) technique, and the diagonalization technique. With a suitable parameter $$\alpha $$ , the WR technique generates a functional sequence $$\{u^k(t)\}$$ via the dynamic iterations $${\dot{u}}^k=f(t,u^k)$$ , $$u^k(0)=\alpha u^k(T)-\alpha u^{k-1}(T)+u_0$$ , and at convergence we get $$u^{\infty }(t)=u(t)$$ . Each WR iterate represents a periodic-like differential equation, which is very suitable for applying the diagonalization technique yielding direct parallel-in-time computation. The parameter $$\alpha $$ controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the WR iterations, and we perform a detailed analysis for the influence of the parameter $$\alpha $$ on the method. We show that the roundoff error is proportional to $$\epsilon (2N+1)\max \{|\alpha |^2, |\alpha |^{-2}\}$$ ( $$N=T/\varDelta t$$ and $$\epsilon $$ is the machine precision), and the convergence factor can be bounded by $$|\alpha |e^{-TL}/(1-|\alpha |e^{-TL})$$ , where $$L\ge 0$$ is the one-sided Lipschitz constant of f. We also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored. Our analysis includes the heat and wave equations as special cases. Numerical results are given to support our findings.

An Efficient Model of Transient Electromagnetic Field Coupling to Multiconductor Transmission Lines Based on Analytical Iterative Technique in Time Domain

Due to the rapid increase of electromagnetic interference in modern society, modeling and analyzing the signal integrity with multiconductor transmission lines (MTLs) has become more and more important. This paper provides an efficient time-domain solution for the transient electromagnetic field coupling to lossless MTLs based on the distributed analytical representation and iterative technique. This method is entirely based on time-domain and can handle the case with nonlinear load conveniently. In this method, the interaction effects between the lines are equivalent to the distributed source along the adjacent lines, and the time-domain analytical expressions of the transient response along transmission lines are derived step by step by adopting the waveform relaxation and iteration technique. This method has a high computational efficiency because it handles the MTLs in a single-line way and does not require to disperse the line. The validation examples show that the proposed method can handle the nonlinear load case with an acceptable accuracy. Furthermore, it has a high computational efficiency especially in the case with a large number of lines.

A unified framework for spiking and gap-junction interactions in distributed neuronal network simulations.

Contemporary simulators for networks of point and few-compartment model neurons come with a plethora of ready-to-use neuron and synapse models and support complex network topologies. Recent technological advancements have broadened the spectrum of application further to the efficient simulation of brain-scale networks on supercomputers. In distributed network simulations the amount of spike data that accrues per millisecond and process is typically low, such that a common optimization strategy is to communicate spikes at relatively long intervals, where the upper limit is given by the shortest synaptic transmission delay in the network. This approach is well-suited for simulations that employ only chemical synapses but it has so far impeded the incorporation of gap-junction models, which require instantaneous neuronal interactions. Here, we present a numerical algorithm based on a waveform-relaxation technique which allows for network simulations with gap junctions in a way that is compatible with the delayed communication strategy. Using a reference implementation in the NEST simulator, we demonstrate that the algorithm and the required data structures can be smoothly integrated with existing code such that they complement the infrastructure for spiking connections. To show that the unified framework for gap-junction and spiking interactions achieves high performance and delivers high accuracy in the presence of gap junctions, we present benchmarks for workstations, clusters, and supercomputers. Finally, we discuss limitations of the novel technology.

Experimental validation of waveform relaxation technique for power system controller testing

A Waveform Relaxation (WR) based iterative real-time playback scheme for controller testing was recently proposed in the literature along with proof-of-concept simulations. This scheme can be a low-cost alternative to Hardware-in-Loop simulation, as it does not require a real-time simulator. To demonstrate practical feasibility of the WR scheme, this paper presents results of experiments with real-time implementation of controllers, iteratively interacting with simulated models of power apparatus via storage and real-time play-back. Two systems are considered: a HVDC controller tested with a detailed model of the converters, and a TCSC based damping controller tested with a low frequency model of a power system. The results are validated with those obtained using simulated models of the controllers. We also present results of an experiment in which the tested HVDC controller is used to control the scaled real-life HVDC apparatus, for which a simulated model was used during controller testing. Convergence and a good match between simulated and real-time implementation are obtained for the HVDC system. The experiments on the TCSC damping controller drawn our attention to a potential convergence problem which may arise due to iteration-dependent round-off noise.

Optimization of Transmission Conditions in Waveform Relaxation Techniques for RC Circuits

Waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors. A limiting factor for classical waveform relaxation, however, is the convergence speed for an important class of problems, especially if long time windows are considered. In contrast, the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem. Today several numerical analyses have shown that optimized waveform relaxation algorithms can overcome slow convergence over long time windows. However, the optimized waveform relaxation techniques require the determination of optimized parameters. In this paper, we present a theoretical foundation for the determination of the optimized parameters for an important class of RC circuits.

Accelerated Simulation of High-Fidelity Models of Supercapacitors Using Waveform Relaxation Techniques

The waveform relaxation (WR) technique is used to accelerate the time-domain simulation of high-fidelity models of supercapacitors. Because of their high power density, supercapacitors are suitable energy storage options in electrified transportation fleets and renewable energy systems. Given their fast charging/discharging profile, high-fidelity models, such as transmission-line models or multistage ladder structures, are conventionally adopted for design and simulation purposes. High-fidelity models, that include fast dynamic modes, can slow down the simulation process. This is problematic, in particular, since supercapacitors are parts of a larger, more complex system, e.g., a power train, with a wide dynamic range. In this paper, frequency-domain characterization of a supercapacitor is conducted by the electrochemical impedance spectroscopy. Then, the equivalent multistage ladder model of the supercapacitor is extracted and parameterized. The high-fidelity model is verified by considering the measured charging profile of the supercapacitor. The WR technique, including circuit partitioning and time windowing, is considered for the resulting high-fidelity model. WR results in an order-of-magnitude improvement in simulation speed while maintaining an excellent agreement with the hardware measurement and conventional simulations.

A mathematical analysis of optimized waveform relaxation for a small RC circuit

Waveform relaxation techniques are an important tool for the simulation of very large scale circuits. They are based on a partition of the circuit into sub-circuits, and then use an iteration between sub-circuits to converge to the solution of the entire circuit. Their importance has increased with the wide availability of parallel computers with a large number of processors. Unfortunately classical waveform relaxation is hampered by slow convergence, but this can be addressed by better transmission conditions, which led to the new class of optimized waveform relaxation methods. In these methods, both voltage and current information is exchanged in a combination which can be optimized for the performance of the method. We prove in this paper a conjecture for the optimal combination for the particular case of a small RC circuit, and also present and analyze a transmission condition which includes a time derivative.

A parareal algorithm based on waveform relaxation

We report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (WR) technique with the classical parareal algorithm. The parallelism can be simultaneously exploited in both sub-systems by WR and time by parareal. We also provide a sharp estimation on errors for the new algorithm. The iterations of parareal and WR are balanced to optimize the performance of the algorithm. Furthermore, the parallel speedup and efficiency of the new approach are analyzed by comparing with the classical parareal algorithm and the WR technique, respectively. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations

We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

Parallel and Scalable Transient Simulator for Power Grids via Waveform Relaxation (PTS-PWR)

This paper presents a fast algorithm for transient simulation of power grids in very large scale integration systems using waveform relaxation (WR) techniques. Novel partitioning methods and convergence accelerators are developed for fast convergence of WR iterations when applied to power grid networks. Unlike the direct solvers, the new method is highly parallelizable and scales well with the increasing number of CPUs, leading to significant speed-ups. Numerical examples are presented to demonstrate the validity and efficiency of the proposed method.

Parallel Simulation of Massively Coupled Interconnect Networks

In a system containing high-speed interconnects, the presence of a large number of coupled lines seriously limits the ability to perform fast simulations. In this paper, a parallel algorithm is presented that allows for simulations of massively coupled interconnects to be performed efficiently. New methods based on physical and time-domain partitioning are developed to create parallelism during transient simulations of large coupled interconnects. In addition, the proposed method exploits the recently developed waveform relaxation techniques to decouple and parallelize the large coupled simulation problem. In this approach, for a simulation of nL lines run on nP processors, the computational complexity is O(nLnP -1). This provides considerable savings as opposed to O(nL s ), 3 ? s ? 4 for full coupled-line simulations.

On numerical methods for highly oscillatory problems in circuit simulation

PurposeThe purpose of this paper is to analyse a novel technique for an efficient numerical approximation of systems of highly oscillatory ordinary differential equations (ODEs) that arise in electronic systems subject to modulated signals.Design/methodology/approachThe paper combines a Filon‐type method with waveform relaxation techniques for nonlinear systems of ODEs.FindingsThe analysis includes numerical examples to compare with traditional methods such as the trapezoidal rule and Runge‐Kutta methods. This comparison shows that the proposed approach can be very effective when dealing with systems of highly oscillatory differential equations.Research limitations/implicationsThe present paper constitutes a preliminary study of Filon‐type methods applied to highly oscillatory ODEs in the context of electronic systems, and it is a starting point for future research that will address more general cases.Originality/valueThe proposed method makes use of novel and recent techniques in the area of highly oscillatory problems, and it proves to be particularly useful in cases where standard methods become expensive to implement.

Applications of Multilinear and Waveform Relaxation Methods for Efficient Simulation of Interconnect-Dominated Nonlinear Networks

Applications of multilinear and waveform relaxation methods are presented for efficient transient analysis of interconnect-dominated nonlinear networks. In this paper, two procedures that realize these well-known and fundamental theories in conventional circuit simulation tools are developed by taking advantage of the unique characteristics of interconnect networks. The multilinear theory uses the Volterra functional series to decompose the nonlinear network into multiple linear networks. Then, the solutions of the mildly nonlinear network are obtained from the linear combinations of sequences of responses of the decomposed linear networks. On the other hand, the waveform relaxation technique is used to solve networks with strong nonlinearity. The networks are partitioned into linear and nonlinear subnetworks and each subnetwork is solved iteratively using the waveform relaxation technique. Simplified analysis steps that give good insight into these techniques are also derived analytically. Finally, the accuracy and efficiency of the methods are verified with two examples.

Waveform Relaxation Techniques for Simulation of Coupled Interconnects With Frequency-Dependent Parameters

The large number of coupled lines in an interconnect structure is a serious limiting factor in simulating high-speed circuits. Waveform relaxation based on transverse partitioning has been previously presented to address this problem for interconnects with constant per-unit-length parameters. This paper extends the waveform relaxation technique to handle the more difficult and important case of frequency-dependent parameters. The computational cost of the proposed algorithm grows linearly with the number of coupled lines

Simulation of Coupled Interconnects Using Waveform Relaxation and Transverse Partitioning

The large number of coupled lines in an interconnect structure is a serious limiting factor in simulating high-speed circuits. A new method is presented for efficient simulation of large interconnects based on transverse partitioning and waveform relaxation techniques. The computational cost of the proposed algorithm grows linearly with the number of coupled lines. In addition, the algorithm is highly suitable for parallel implementation leading to further significant reduction in the computational complexity.

Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs

Advanced time discretization schemes for stiff systems of ordinary differential equations (ODEs), such as implicit Runge--Kutta and boundary value methods, have many appealing properties. However, the resulting systems of equations can be quite large and expensive to solve. Many techniques, exploiting the structure of these systems, have been developed for general ODEs. For spatial discretizations of time-dependent partial differential equations (PDEs) these techniques are in general not sufficient and also the structure arising from spatial discretization has to be taken into consideration. We show here that for time-dependent parabolic problems, this can be done by multigrid methods, as in the stationary elliptic case. The key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously. The overall cost is essentially proportional to the cost of integrating a scalar ODE for each grid point. Combination of the multigrid principle with both time stepping and waveform relaxation techniques is described, together with a convergence analysis. Numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients.