Explicit Multistep Method Research Articles

FPGA Implementation of Sliding Mode Control and Proportional-Integral-Derivative Controllers for a DC–DC Buck Converter

DC–DC buck converters have been designed by incorporating different control stages to drive the switches. Among the most commonly used controllers, the sliding mode control (SMC) and proportional-integral-derivative (PID) controller have shown advantages in accomplishing fast slew rate, reducing settling time and mitigating overshoot. The proposed work introduces the implementation of both SMC and PID controllers by using the field-programmable gate array (FPGA) device. The FPGA is chosen to exploit its main advantage for fast verification and prototyping of the controllers. In this manner, a DC–DC buck converter is emulated on an FPGA by applying an explicit multi-step numerical method. The SMC controller is synthesized into the FPGA by using a signum function, and the PID is synthesized by applying the difference quotient method to approximate the derivative action, and the second-order Adams–Bashforth method to approximate the integral action. The FPGA synthesis of the converter and controllers is performed by designing digital blocks using computer arithmetic of 32 and 64 bits, in fixed-point format. The experimental results are shown on an oscilloscope by using a digital-to-analog converter to observe the voltage regulation generated by the SMC and PID controllers on the DC–DC buck converter.

Numerical method in solving neutral and retarded Volterra delay integro-differential equations

The aim of this research is to produce accurate numerical results in solving neutral Volterra delay integro-differential equations (NVDIDE) and retarded Volterra delay integro-differential equations (RVDIDE) of constant type. A third-order explicit multistep block method is derived by applying the Taylor series. The consistency, zero stability, and convergence of the method are determined. The problems are solved by approximating two points simultaneously with constant step size. The delay arguments are approximated using previously calculated values while the integration part is approximated using the quadrature rule. The numerical results obtained have shown that the proposed explicit method is comparable with the other methods and is assumed to be reliable in solving NVDIDE and RVDIDE of constant type.

Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

In this paper, implicit–explicit multistep Galerkin methods are studied for two-dimensional nonlinear Schrödinger equations and coupled nonlinear Schrödinger equations. The spatial discretization is based on Galerkin method using linear and quadratic basis functions on triangular and rectangular finite elements. And the implicit–explicit multistep method is used for temporal discretization. Linear and nonlinear numerical tests are presented to verify the validity and efficiency of the numerical methods. The numerical results record that the optimal order of the error in L2 and L∞ norm can be reached.

Multistep Method of the Numerical Solution of the Problem of Modeling the Circulation of Atmosphere in the Cauchy Problem

An explicit multistep one-stage method is considered, which allows numerical integration of the differential equations that constitute the basis of the atmosphere circulation model by transforming the initial---boundary-value convection---diffusion problem to the Cauchy problem. The method has an advantage over the available methods due to its high precision and low computational cost.

Numerical study using explicit multistep Galerkin finite element method for the MRLW equation

In this article, an explicit multistep Galerkin finite element method for the modified regularized long wave equation is studied. The discretization of this equation in space is by linear finite elements, and the time discretization is based on explicit multistep schemes. Stability analysis and error estimates of our numerical scheme are derived. Numerical experiments indicate the validation of the scheme by L2– and L∞– error norms and three invariants of motion.4 © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1875–1889, 2015

Precise Integration Method of Pseudo-Dynamic Testing of Structures

Based on nonlinear precise integration method, two new numerical integration methods of pseudo-dynamic test of structures are presented. One is explicit predict-correct, four order accuracy and multistep method avoiding calculating the inversion of the state matrix. The other is implicit predict-correct, four order accuracy and multistep method that need calculate the inversion of the state matrix. Since their accuracies are superior to the central difference method by enlarging time step and their stabilities are better, the structural systems of multi-degree of freedom could be well tested and the testing work would be largely reduced. Finally, a pseudo-dynamic test of combined tube structure has been executed with the explicit multi-step method.

Explicit Multistep Mixed Finite Element Method for RLW Equation

An explicit multistep mixed finite element method is proposed and discussed for regularized long wave (RLW) equation. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is discretized by the explicit multistep method. The optimal error estimates in <svg style="vertical-align:-0.0pt;width:16.1625px;" id="M1" height="15.8375" version="1.1" viewBox="0 0 16.1625 15.8375" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> and <svg style="vertical-align:-0.0pt;width:21.362499px;" id="M2" height="15.8375" version="1.1" viewBox="0 0 21.362499 15.8375" width="21.362499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> <g transform="matrix(.012,-0,0,-.012,14.975,7.613)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg> norms for the scalar unknown <svg style="vertical-align:-0.1638pt;width:9.2749996px;" id="M3" height="7.9499998" version="1.1" viewBox="0 0 9.2749996 7.9499998" width="9.2749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D462" d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74&#xA;t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" /></g> </svg> and its flux <svg style="vertical-align:-3.50804pt;width:44.3125px;" id="M4" height="12.1125" version="1.1" viewBox="0 0 44.3125 12.1125" width="44.3125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" /></g><g transform="matrix(.017,-0,0,-.017,13.118,7.675)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,27.821,7.675)"><use xlink:href="#x1D462"/></g> <g transform="matrix(.012,-0,0,-.012,36.975,11.762)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g> </svg> based on time explicit multistep method are derived. Some numerical results are given to verify our theoretical analysis and illustrate the efficiency of our method.

Explicit multistep method for the numerical solution of RLW equation

In this paper, we present explicit multistep method for regularized long wave (RLW) equation. The discretization in space is based on the Galerkin method using linear space finite elements, and for the time discretization the explicit multistep method is used. Propagation of solitary waves and interaction of two solitary waves are simulated using the proposed method to confirm the theoretical results.

Explicit multistep method for the numerical solution of stiff differential equations

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.

A fast and stable method for Raman amplifier propagation equations

A novel predictor-corrector method for the coupled equations for ultrabroad-band Raman amplifiers with multiple pumps is proposed and derived, for the first time, based on the Adams formula. The proposed algorithm is effective in solving Raman amplifier equations that include pumps, signals, noises, and their backscattering waves. The detail procedure is given, and proves the excellence of our algorithm. Simulation results show that, in designing the Raman amplifier, our multistep method can effectively improve the accuracy and stability compared with the one-step method and explicit multistep method. The numerical results show that the power of backscattering pumps and signals is lower by ~30 dB and 20 dB than their original power, respectively, and the power of forward and backward noises is less than that of input signals by ~30 dB under our simulation conditions.

Predictor-Corrector Algorithms with Identical Regions of Stability

It is shown that the general $PEC$ algorithm has a region of stability (absolute or relative) identical with that of a certain explicit linear multistep method $\tilde P$, whose order can be simply expressed in terms of the orders of the original predictor and corrector. Further, the region of stability of the general $P(EC)^{m + 1} $ algorithm is shown to be identical with that of the algorithm $\tilde P(EC)^m E$. It is concluded that there is little motivation for using $PEC$ algorithms, and that, among predictor-corrector algorithms employing $m + 1$ evaluations, the “best” algorithms, from the point of view of order and stability, will be found among $P(EC)^m E$ algorithms; the argument, however, takes no account of step number.