A new approach to estimating a numerical solution in the error embedded correction framework

Error estimation using neural network technique for solving ordinary differential equations

“…For this reason, several numerical techniques for solving ODEs have been developed during last few decades. Also, the numerical methods can be broadly classified into the following categories: the first class consists of one-step multistage techniques such as Runge-Kutta-type methods [5,13,17], the second includes BDF-type multistep methods [6], and the last is a group of deferred or error correction methods [4,7,18,19] such as spectral deferred correction (SDC) methods [8,11], etc.…”

“…Unlike the traditional numerical schemes, the deferred or error correction schemes are investigated for the numerical errors with a provisional solution which is preferentially calculated by any numerical scheme. It is already shown that these schemes can have higher convergence order and higher accuracy without any loss of stability [4,7,18,19,29].…”

Stability analysis of a diagonally implicit scheme of block backward differentiation formula for stiff pharmacokinetics models

“…The method with a full coefficient matrix, i.e., FIRK requires to solve a system of (n-dimensional × r-stages) nonlinear equations in each of its integration stages [8]. To reduce the computational cost of evaluating the stages in the FIRK method, [7][8][9][10][11] opted for the DIRK method. As stated in [6], this method is characterized by a lower triangular A-matrix with a ij = 0 for i < j and is sometimes referred to as semi-implicit or semi-explicit Runge-Kutta methods.…”

Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

“…Considering the impact, a new perspective to compare the potentials of both methods should be investigated as well as existing comparative studies. First of all, it is well known that the highest order of an A-stable multi-step method is two, so lots of research [12][13][14][15][16][17][18][19][20][21][22][23][24] developing higher order methods have focused on either multi-step methods satisfying some less restrictive stability condition or multi-stage methods which combine A-stability with high-order accuracy [2,[25][26][27][28][29]. In addition, multi-stage methods such as Runge-Kutta (RK) type methods do not require any additional memory for function values at previous steps since it does not use any previously computed values [30][31][32].…”