An Optimized Symmetric 8-Step Semi-Embedded Predictor-Corrector Method for IVPs with Oscillating Solutions

Transcription

1 Appl. Math. Inf. Sci., No., - () Applied Mathematics & Information Sciences An International Journal An Optimized Symmetric -Step Semi-Embedded Predictor-Corrector Method for IVPs with Oscillating Solutions G. A. Panopoulos and T. E. Simos, Laboratory of Computational Sciences, Department of Computer Science and Technology,Faculty of Sciences and Technology, University of Peloponnese, GR- Tripolis, GREECE Department of Mathematics, College of Sciences, King Saud University, P. O. Box, Riyadh, KSA Received: Jun ; Revised Sep. ; Accepted Sep. Published online: Jan. Abstract: In this paper we present a new optimized symmetric eight-step semi-embedded predictor-corrector method (SEPCM) with minimal phase-lag. The method is based on the symmetric multistep method of Quinlan-Tremaine [], with eight steps and eighth algebraic order and is constructed to solve IVPs with oscillating solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved. Keywords: Orbital problems, phase-lag, initial value problems, oscillating solution,symmetric, multistep, predictor-corrector, semiembedded.. Introduction We study the numerical integration of special second-order periodic initial-value problems of the form y (x) =f(x, y), y(x )=y and y (x )=y () with an oscillatory solutions. These ordinary differential equations are of second order in which the derivative y does not appear explicitly (see for numerical methods for these problems [] - [9] and references therein).. Phase-lag analysis of symmetric multistep methods For the numerical solution of the above initial value problem (), multistep methods of the form m a i y n+i = h m b i f(x n+i,y n+i ) () with m steps can be used over the equally spaced intervals {x i } m [a, b] and h = x i+ x i, i = ()m. If the method is symmetric then a i = a mi and b i = b mi, i = () m. Method () is associated with the operator L(x) = m a i u(x + ih) h where u C. m b i u (x + ih) () Definition.The multistep method () is called algebraic of order p if the associated linear operator L vanishes for any linear combination of the linearly independent functions, x,x,..., x p+. When a symmetric k-step method, that is for i = k()k, is applied to the scalar test equation y = ω y () Corresponding author: tsimos.conf@gmail.com

2 G. A. Panopoulos, T. E. Simos : Optimized Symmetric -Step Semi-Embedded Predictor-Corrector... a difference equation of the form A k (v)y n+k A (v)y n+ + A (v)y n +A (v)y n A k (v)y nk = () is obtained, where v = ωh, h is the step length and A (v), A (v),..., A k (v) are polynomials of v. The characteristic equation associated with () is A k (v)s k A (v)s + A (v) +A (v)s A k (v)s k = () From Lambert and Watson (9) we have the following definitions: Definition.A symmetric k-step method with characteristic equation given by () is said to have an interval of periodicity (,v ) if, for all v (,v ), the roots s i,i = ()k of Eq. () satisfy: s = e iθ(v),s = e iθ(v),and s i, i= ()k () where θ(v) is a real function of v. Definition.For any method corresponding to the characteristic equation () the phase-lag is defined as the leading term in the expansion of t = v θ(v) () Then if the quantity t = O(v q+ ) as v, the order of phase-lag is q. Theorem.[] The symmetric k-step method with characteristic equation given by () has phase-lag order q and phase-lag constant c given by cv q+ + O(v q+ )=(9) A k (v)cos(kv)+... +A j (v)cos(jv) A (v) k A k (v)+... +j A j (v)+... +A (v) The formula proposed from the above theorem gives us a direct method to calculate the phase-lag of any symmetric k- step method. In our case, the symmetric -step method has phase-lag order q and phase-lag constant c given by: cv q+ + O(v q+ )= T T () T =A (v) cos(v)+a (v) cos(v) + A (v) cos(v)+a (v)cos(v)+a (v) T =A (v)+a (v)+a (v)+a (v). Construction of the new optimized semi-embedded predictor-corrector method From the form () and without loss of generality we assume a m =and we can write m y n+m + a i y n+i = h finally we get m y n+m = a i y n+i + h m m b i f(x n+i,y n+i ), () b i f(x n+i,y n+i ) () If the method is symmetric then a i = a mi and b i = b mi, i = () m... The explicit method phase-lag order infinite (phase-fitted) From the form () with m =and b =we get the form of the explicit symmetric eight-step methods: ( y = y + a (y + y )+a (y + y ) +a (y + y )+a y ) + h ( b (f + f ) +b (f + f )+b (f + f )+b f ). () The characteristic equation () becomes A (v)s + A (v)s + A (v)s + A (v)s + A (v) +A (v)s + A (v)s + A (v)s + A (v)s = () where A i (v) =a i + v b i,i= (), a =,b =. () From () with a =, a =,a =, a =, b = 9,b = 9,b = 9 9,b = 9, y i = y(x + ih) and f i = f(x + ih, y(x + ih)), where h is the step length, we obtain the symmetric multistep method of Quinlan- Tremaine [], with eight steps and eighth algebraic order. This method has an interval of periodicity (,v ) where v =..

3 Appl. Math. Inf. Sci., No., - () / From () and by keeping the same a i coefficients and by nullifying the phase-lag, we get: a =, a =,a =, a =, b = b +,b =b, b = b + 9, b = 9 v (cos (v) ) ( 9 (cos (v)) + 9 (cos (v)) + ( 9 v ) (cos (v)) + ( + v ) ) cos (v) v + () y i = y(x+ih), f i = f(x+ih, y(x+ih)) where v = ωh, ω is the frequency and h is the step length. For small values of v the above formulae are subject to heavy cancelations. In this case the following Taylor series expansion must be used: b = 9 v + 9 v 9 v v 99 9 v 9 v 9 v 9 9 v 999 v v,() where v = ωh, ω is the frequency and h is the step length. The explicit symmetric multistep method () has an interval of periodicity (,v ) where v =.. In order to find the Local Truncation Error(LTE), we express y ±i,i= () and f ±j,j= () via Taylor series and we substitute in (). Based on this procedure we obtain the following expansion for the LTE: L.T.E. = h ( y () n + y () n ω ) + O(h ) () The above optimized explicit symmetric multistep method () has eight steps, eight algebraic order and infinite order of phase-lag (phase-fitted) (see [])... The implicit method (corrector) From () for m =, we get the form of the symmetric implicit eight-step method: ( y = y + α (y + y )+α (y + y ) ) +α (y + y )+α y +h ( β (f + f )+β (f + f ) +β (f + f )+β (f + f )+β f ). (9) The characteristic equation () becomes A (v)s + A (v)s + A (v)s + A (v)s + A (v) +A (v)s + A (v)s + A (v)s + A (v)s = () where A i (v) =α i + v β i, i = (), α =. From (9) and by keeping the same a i coefficients (α i = a i,i = (), where a =,a =, a =,a =, a = ) we satisfy as many algebraic equations as possible. After achieving th algebraic order we obtain the implicit symmetric multistep method: α =,α =, α =,α =, α = β =,β = 99,β = 9, β =,β = () with eight steps and tenth algebraic order. The implicit symmetric multistep method () has an interval of periodicity (,v ) where v =.9. The local truncation error of the above method is given by: L.T.E. = 9 h y n () + O(h ) () We note that the symmetric multistep method of Quinlan- Tremaine [], with eight steps and eighth algebraic order, derives from () if we take β =(see [])... The new predictor-corrector method From J.D. Lambert (99) we have that the general k-step predictor-corrector or PC pair is: m j= j= a j y n+j = h m j= j= b j f n+j m a j y n+j = h m b j f n+j ()

4 G. A. Panopoulos, T. E. Simos : Optimized Symmetric -Step Semi-Embedded Predictor-Corrector... Let the predictor and corrector defined by () have orders p and p respectively. The order of a PC method depend on the gap between p and p and on λ, the number of times the corrector is called. If p <p and λ =< p p, the order of the PC method is p + λ(< p) []. We consider the pair of linear multistep methods: m a i y n+i = h m b i (v)f(x n+i,y n+i ) m a i y n+i = h m β i f(x n+i,y n+i ) where a + b (v), a + β, v = ωh, ω is the frequency and h is the step length. Without loss of generality we assume a m =and we can write y n+m + m y n+m + m and we have a i y n+i = h m + m y n+m = m y n+m = m b i (v)f(x n+i,y n+i ) a i y n+i = h (β m f(x n+m,y n+m ) ) β i f(x n+i,y n+i ) a i y n+i + h m +h m b i f(x n+i,y n+i ) a i y n+i + h β m f(x n+m,y n+m ) β i f(x n+i,y n+i ) If we call A n = m a i y n+i, we can write y n+m = A n + h m b i (v)f(x n+i,y n+i ) y n+m = A n + h β m f(x n+m,y n+m ) +h m β i f(x n+i,y n+i ) From this pair, a new predictor-corrector ( PC ) pair form, is formally defined as follows: yn+m = A n + h m b i (v)f(x n+i,y n+i ) y n+m = A n + h β m f(x n+m,yn+m) +h m β i f(x n+i,y n+i ) () where A n = m a i y n+i, a + b (v), a + β, v = ωh, ω is the frequency and h is the step length. We call the above method Semi-Embedded Predictor-Corrector Method (SEPCM), in the sense that a part of the predictor method is contained in the corrector method. If the method is symmetric then a i = a mi, b i (v) = b mi (v) and β i = β mi, i = () m. From (), () and () a new symmetric eight-step semi-embedded predictor-corrector method (SEPCM) with minimal phase-lag obtained: A = y +(y + y ) (y + y )+(y + y ) y = A + h (b (v)(f + f )+ ( 9 b (v) ) (f + f )+ ( ) b (v) (f + f + ( b (v) ) ) f ( y = A + h (f + f )+ (f + f ) 9 (f + f )+ 99 (f + f )+ f () where T b (v) = 9 v (cos (v) ), () where T = 9 (cos (v)) +9 (cos (v)) + ( 9 v ) (cos (v)) + ( + v ) cos (v) v +and y i = y(x + ih), f i = f(x + ih, y(x + ih)), v = ωh, ω is the frequency and h is the step length. For small values of v the following Taylor series expansions must be used: b (v) = 9 v + 9 v 9 v v 99 v 9 9 v 9 v 9 9 v 999 v v, () where v = ωh, ω is the frequency and h is the step length. The characteristic equation () becomes A (v)s + A (v)s + A (v)s + A (v)s + A (v) +A (v)s + A (v)s + A (v)s + A (v)s = ()

5 Appl. Math. Inf. Sci., No., - () / where A i (v) =α i + v (β i a i β ) v b i (v) β, i = (), α = a =, b (v) =. (9) The new optimized symmetric eight-step semi-embedded predictor-corrector method () has an interval of periodicity (,v ) where v =.. The local truncation error of the above method is given by: L.T.E. = ( 9 9 y() n + 99 y() n ω ) h + O(h ) () The new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) () has eight steps, tenth algebraic order and tenth order of phase-lag.. Numerical results... Orbital Problem by Stiefel and Bettis The almost periodic orbital problem studied by [] can be described by y + y =. e ix, y() =, y () =.999 i, y C, () or equivalently by u + u =. cos(x), u() =, u () =, v + v =. sin(x), v() =, v () =.999. () The theoretical solution of the problem () is given below: y(x) =u(x)+iv(x), u,v R u(x) = cos(x)+. x sin(x), v(x) =sin(x). x cos(x). The system of equations () has been solved for x [, π]. Estimated frequency: w =... The problems... Two-dimensional Kepler problem (Two-Body Problem) The efficiency of the new optimized symmetric semi-embedded eight-step predictor-corrector method will be measured through the integration of four initial value problems with oscillating solution. + z ) (y + z ) y y =,z z =, () (y with y() = e, y () =, z() =, z () =... Duffing Equation y = y y +. cos(. t), with y() =.,y () =, t [, π]. Theoretical solution: +e e,t [, π], where e is the eccentricity. The theoretical solution of this problem is given below: y(t) =cos(u) e z(t) = e sin(u). where u can be found by solving the equation ue sin(u) t =. y(t) =.9 cos(. t) +.9 cos(. t) +. cos(. t) +. cos(. t)+... () Estimated frequency: w =.... Nonlinear Equation y = y +sin(y), with y() =,y () =, t [, π]. The theoretical solution is not known, but we use y( π) =.999. Estimated frequency: w =. We used the estimation w = (y +z ) the problem... The methods as frequency of We have used several multistep methods for the integration of the four test problems. These are: The new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order and minimal phase-lag () (New SEPCM ste) The symmetric -step method of Quinlan-Tremaine of order ten [] (QT) The optimized symmetric -step method () of order eight [] (QT Phase-fitted)

6 G. A. Panopoulos, T. E. Simos : Optimized Symmetric -Step Semi-Embedded Predictor-Corrector... The symmetric -step method of Quinlan-Tremaine of order eight [] (QT) The -stage exponentially-fitted method of Simos and Aguiar of algebraic order nine [9] (SA Ste EF) The symmetric -step method of Jenkins of order six [] (Jenkins ste) The -step, -stage exponentially-fitted predictor-corrector method (EF) of Psihoyios and Simos of algebraic order five [] (PS Ste EF) The -step, -stage exponentially-fitted predictor-corrector method of Simos and Williams of algebraic order six [] (SW Ste Sta EF PC) The -step predictor-corrector method Adams-Bashforth - Moulton of order four () The -step predictor-corrector method Milne-Simpson of order four ()... Comparison We present the accuracy of the tested methods expressed by the log (max. error over interval) or log (error at the end point), depending on whether we know the theoretical solution or not, versus the CPU time. In Table we see the comparison of the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) () and the multistep symmetric method of Quinlan-Tremaine [] with eight steps for all the problems solved. In Figure we see the results for the Duffing equation, in Figure the results for the Nonlinear equation, in Figure the results for the Stiefel-Bettis almost periodic problem, and in Figures and the results for the twodimensional Kepler problem (Two-body problem) for eccentricities e=. and e=.. Among all the methods used, the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order is the most efficient. The interval of periodicity of the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order is about. times larger than the multistep symmetric method of Quinlan-Tremaine with eight steps and eighth algebraic order and about two times larger than the optimized symmetric -step method () with eight steps and eighth algebraic order. The new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order can achieve the required accuracy with a step-size eight times larger than the multistep symmetric method of Quinlan-Tremaine with eight steps and eighth algebraic order for the Duffing equation and with a step-size two times larger than the multistep symmetric method of Quinlan- Tremaine with eight steps and eighth algebraic order for all the other problems solved. An interesting remark is that the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order, is more efficient than the multistep symmetric method of Quinlan-Tremaine, with ten steps and tenth algebraic order.. Conclusions We have constructed a new predictor-corrector ( PC ) pair form for the numerical integration of second-order initialvalue problems (), for which we know the frequency. From the form () we have developed a new optimized eightstep symmetric semi-embedded predictor-corrector method (SEPCM) (). The new method has algebraic order ten and it can be used to solve numerically, related initial-value problems with oscillatory solutions, for which we know the frequency. The new semi-embedded predictor-corrector pair form () has the advantage that reduces the computational expense if the additions on the factor m a i y n+i, are done twice. We have applied the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order () along with a group of several methods from the literature to four oscillatory problems. We concluded that that the new optimized eight-step symmetric semi-embedded predictor-corrector method (SEPCM) with tenth algebraic order () is highly efficient compared to other methods. TABLE. Comparison of the new SEPCM of new step and Quinlan Quinlan Trimaine step method for all the problems solved the problems solved Test Problem Method (digits) CPU Time Step Length Maximum Error Duffing's Equation Nonlinear Equation Stiefel Bettis Two Body Problem Eccentricity. Eccentricity. Quinlan Tremaine step E New SEPCM step e Quinlan Tremaine step e New SEPCM step.9.9..e Quinlan Tremaine step.9...e New SEPCM step E Quinlan Tremaine step e 9 New SEPCM step....9e 9 Quinlan Tremaine step..9..e New SEPCM step E Table Comparison for CPU time and Step Length References [] D.G. Quinlan and S. Tremaine, Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits, The Astronomical Journal,,, 9- (99) [] J.M. Franco, M. Palacios, JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS,, (99) [] J.D.Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem, Pages -, John Wiley and Sons. (99)

7 Appl. Math. Inf. Sci., No., - () / 9 Duffing's Equation New SEPCM ste 9 QT Phase fitted QT QT SA Ste EF Jenkins ste PS Ste EF 9 New SEPCM ste QT Phase fitted QT QT SA Ste EF Jenkins ste PS Ste EF Two - Body Problem (e=.) Figure Efficiency for the Duffing equation Figure Efficiency for the Two-body problem using eccentricity e =. 9 Nonlinear 9 Figure Efficiency for the Nonlinear equation 9 Stiefel - Bettis New SEPCM ste QT Phase fitted QT QT SA Ste EF Jenkins ste PS Ste EF New SEPCM ste QT Phase fitted QT QT SA Ste EF Jenkins ste PS Ste EF 9 Figure Efficiency for the Orbital Problem by Stiefel and Bettis [] E. Stiefel, D.G. Bettis, Stabilization of Cowell s method, Numer. Math., - (99) [] G.A. Panopoulos, Z.A. Anastassi and T.E. Simos: Two New Optimized Eight-Step Symmetric Methods for the Efficient Solution of the Schrödinger Equation and Related Problems, MATCH Commun. Math. Comput. Chem.,,, - () [] G.A. Panopoulos, Z.A. Anastassi and T.E. Simos, Two optimized symmetric eight-step implicit methods for initialvalue problems with oscillating solutions, Journal of Mathematical Chemistry () -(9) New SEPCM ste QT Phase fitted QT QT SA Ste EF Jenkins ste PS Ste EF Two - Body Problem (e=.) 9 Figure Efficiency for the Two-body problem using eccentricity e =. [] [] T.E. Simos and P.S. Williams, Bessel and Neumann fitted methods for the numerical solution of the radial Schrödinger equation, Computers and Chemistry,, -9 (9) [9] T.E. Simos and Jesus Vigo-Aguiar, A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems, Computer Physics Communications,, -9 () [] T.E. Simos and G. Psihoyios, JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS (): IX-IX MAR [] T. Lyche, Chebyshevian multistep methods for Ordinary Differential Eqations, Num. Math. 9, - (9) [] T.E. Simos, Chemical Modelling - Applications and Theory Vol., Specialist Periodical Reports, The Royal Society of Chemistry, Cambridge () [] J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial values problems, J. Inst. Math. Appl. 9 (9) [] A. Konguetsof and T.E. Simos, A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS () 9-() [] Z. Kalogiratou, T. Monovasilis and T.E. Simos, Symplectic integrators for the numerical solution of the Schrödinger

8 G. A. Panopoulos, T. E. Simos : Optimized Symmetric -Step Semi-Embedded Predictor-Corrector... equation, JOURNAL OF COMPUTATIONAL AND AP- PLIED MATHEMATICS () -9() [] Z. Kalogiratou and T.E. Simos, Newton-Cotes formulae for long-time integration, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS () -() [] G. Psihoyios and T.E. Simos, Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS () -() [] T.E. Simos, I.T. Famelis and C. Tsitouras, Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions, NUMERICAL ALGORITHMS () -() [9] T.E. Simos, Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution, AP- PLIED MATHEMATICS LETTERS () -() [] K. Tselios and T.E. Simos, Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics, JOURNAL OF COMPU- TATIONAL AND APPLIED MATHEMATICS Volume: () -() [] D.P. Sakas and T.E. Simos, Multiderivative methods of eighth algrebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation, JOUR- NAL OF COMPUTATIONAL AND APPLIED MATHE- MATICS () -() [] G. Psihoyios and T.E. Simos, A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions, JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS () - () [] Z. A. Anastassi and T.E. Simos, An optimized Runge-Kutta method for the solution of orbital problems, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS () -9() [] T.E. Simos, Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems, APPLIED MATHEMATICS LETTERS Volume: () -(9) [] S. Stavroyiannis and T.E. Simos, Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs, APPLIED NUMERICAL MATHEMATICS 9() -(9) [] T.E. Simos, Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation, ACTA APPLICANDAE MATHEMATICAE () - () [] T. E. Simos, New Stable Closed Newton-Cotes Trigonometrically Fitted Formulae for Long-Time Integration, Abstract and Applied Analysis, Volume, Article ID, pages, doi:.// [] T.E. Simos, Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag, Journal of Applied Mathematics, Volume, Article ID, pages, doi:.//, [9] Z.A. Anastassi and T.E. Simos, A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 9() G. A. Panopoulos is currently researcher, in the Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Science and Technology, University of Peloponnese, Greece, teaching part of the subjects Numerical Linear Algebra, Numerical Analysis, Operations Research (Linear Programming) and programming in Matlab, in cooperation with Prof. Dr. T.E. Simos. He obtained his PhD in Computational Science and Numerical analysis from University of Peloponnese, Greece. He is President of the Department of Arcadia of the Hellenic Mathematical Society. He has published more than ten research articles in reputed international journals of mathematical, chemistry, physics, astronomy and engineering sciences. Research interests: Computational Science, Numerical analysis, numerical solution of periodic and oscillatory initial-value and boundary-value problems, applications in computational chemistry, in computational physics, in astronomy, scientific computing. Theodore E. Simos (b. 9 in Athens, Greece) is a Visiting Professor within the Distinguished Scientists Fellowship Program at the Department of Mathematics, College of Sciences, King Saud University, P. O. Box, Riyadh, Saudi Arabia and Professor at the Laboratory of Computational Sciences of the Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR- Tripolis, Greece. He holds a Ph.D. on Numerical Analysis (99) from the Department of Mathematics of the National Technical University of Athens, Greece. He is Highly Cited Researcher in Mathematics, Active Member of the European Academy of Sciences and Arts, Active Member of the European Academy of Sciences and Corresponding Member of European Academy of Sciences, Arts and Letters. He is Editor-in-Chief of three scientific journals and editor of more than scientific journals. He is reviewer in several other scientific journals and conferences. His research interests are in numerical analysis and specifically in numerical solution of differential equations, scientific computing and optimization. He is the author of over peer-reviewed publications and he has more than citations (excluding self-citations).