Science Explorer Back to mapMap
Discussion
Topic · Numerical Analysis

Numerical methods for differential equations

Numerical methods for differential equations is a research topic within Numerical Analysis. Science Explorer counts 37k research works in it since 1950. 10.0% of them reached the world's top 10% most cited for their field and year.

This cluster of papers focuses on the development and analysis of numerical integration methods for solving differential equations, with a particular emphasis on exponential integrators, symplectic methods, time-stepping schemes, and variational integrators. The research covers a wide range of applications including the numerical solution of the Schrödinger equation, Hamiltonian systems, and matrix exponential computations.

  • Numerical Integration
  • Differential Equations
  • Exponential Integrators
  • Symplectic Methods
  • Time-Stepping Schemes
  • Variational Integrators
  • Matrix Exponential
  • Runge-Kutta Methods
  • Schrödinger Equation
  • Hamiltonian Systems
Research works
37k
fractional, since 1950
In the world top 10%
3.7k
per year above
Top-10% rate
10.0%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+14%
the tick is no change
Largest producer
China
36% of the 10 largest, 2022-now
Numerical methods for differential equations

Which countries lead Numerical methods for differential equations research?

By volume, China and the United States publish the most (1.2k and 558 works in 2022-now).

By volume, 2022-now

  1. 1 China 1.2k works
  2. 2 United States 558 works
  3. 3 India 351 works
  4. 4 Germany 283 works
  5. 5 Russia 224 works
  6. 6 France 195 works
  7. 7 Italy 161 works
  8. 8 Saudi Arabia 139 works
  9. 9 Türkiye 134 works
  10. 10 Nigeria 132 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 36.3%United States: 16.3%India: 10.3%Germany: 8.3%6 others listed: 28.8%36%largest
China1,240 · 36.3%United States558 · 16.3%India351 · 10.3%Germany283 · 8.3%6 others listed984 · 28.8%

Shares of the rows listed above, not of the whole node.

Which institutions lead Numerical methods for differential equations research?

By volume in 2022-now, Centre National de la Recherche Scientifique publishes the most Numerical methods for differential equations research, followed by Russian Academy of Sciences and Technical University of Munich.

By volume, 2022-now

  1. 1 Centre National de la Recherche ScientifiqueFrance 87 works
  2. 2 Russian Academy of SciencesRussia 75 works
  3. 3 Technical University of MunichGermany 30 works
  4. 4 Chinese Academy of SciencesChina 28 works
  5. 5 Xi'an Jiaotong UniversityChina 28 works
  6. 6 Harbin Institute of TechnologyChina 24 works
  7. 7 Helmholtz Association of German Research CentresGermany 23 works
  8. 8 Shandong UniversityChina 21 works
  9. 9 Huazhong University of Science and TechnologyChina 20 works
  10. 10 Xiangtan UniversityChina 19 works

Who are the leading researchers in Numerical methods for differential equations?

The most-cited researchers publishing on Numerical methods for differential equations include Donald G. Truhlar, Michael I. Jordan and H. Vincent Poor.

  1. 1 Donald G. Truhlar United States 230k citations
  2. 2 Michael I. Jordan United States 182k citations
  3. 3 H. Vincent Poor United States 165k citations

Ranked by citations received across their whole record, among researchers with at least three works on this topic.

What are the most cited papers in Numerical methods for differential equations?

The most cited paper in Numerical methods for differential equations is "Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces" (Rainer Storn, Kenneth V. Price, 1997), with 29,587 citations. Next come "Ordinary Differential Equations" (7,320) and "Theory of Ordinary Differential Equations" (6,268).

Most cited papers
#PaperReleasedCitations
1 Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces Rainer Storn, Kenneth V. Price · Journal of Global Optimization Dec 1997 29,587
2 Ordinary Differential Equations John M. Stewart · Cambridge University Press eBooks Book chapter 12 Oct 2017 7,320
3 Theory of Ordinary Differential Equations Earl A. Coddington, Norman Levinson, Theodor Teichmann · Physics Today Feb 1956 6,268
4 State-space solutions to standard H/sub 2/ and H/sub infinity / control problems John C. Doyle, K. Glover, Pramod P. Khargonekar et al. · IEEE Transactions on Automatic Control 1989 5,567
5 P-LINCS: A Parallel Linear Constraint Solver for Molecular Simulation Berk Hess · Journal of Chemical Theory and Computation Dec 2007 4,177
All 98 most cited papers, with filters and sorting

Where is Numerical methods for differential equations research done?

The largest centres of Numerical methods for differential equations research in 2022-now are Beijing (China), Moscow (Russia), Shanghai (China) and Nanjing (China). Among places with at least 20 works in it, it is an unusually large share of all research in Xiangtan.

Largest cities, 2022-now

  1. 1 BeijingChina 159 works
  2. 2 MoscowRussia 100 works
  3. 3 ShanghaiChina 86 works
  4. 4 NanjingChina 71 works
  5. 5 Xi'anChina 60 works
  6. 6 ParisFrance 57 works
  7. 7 ChangshaChina 54 works
  8. 8 JinanChina 46 works
  9. 9 GuangzhouChina 42 works
  10. 10 WuhanChina 41 works

Where it is the local speciality

  1. XiangtanCN · 24.7 works10×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Numerical methods for differential equations than the world average.

See Numerical methods for differential equations on the map

Where is the best place to study Numerical methods for differential equations?

Among universities, judged by research, Hunan University of Technology, Technical University of Munich and Imam Mohammad ibn Saud Islamic University score highest, combining excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%20%40%60%mean 22.62%fractional works in this field (log) →share of work in the world top 10% →Hunan University of …Vellore Institute of…Imam Mohammad ibn Sa…Technical University…
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Excellenceshare of its work in the world top 10%, 2022-now and all time averaged
40%
Specialisationhow much of its work is in this field, counted up to 10× the world share
10%
Sizeits output in this field, on a log scale
30%
Growthoutput growth, counted from -20% to +20%
10%
Internationalshare of its work with a co-author abroad
10%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreTop 10%SpecialisationWorksGrowth
1Hunan University of TechnologyChina 65.555.5%20.5×10 +138.5%
2Technical University of MunichGermany 58.84.6%7.4×30 +83.9%
3Imam Mohammad ibn Saud Islamic UniversitySaudi Arabia 56.530.3%11.1×9 +192.9%
4Vellore Institute of Technology UniversityIndia 52.131.7%2.7×12 +690.0%
5Huazhong University of Science and TechnologyChina 51.719.2%2.4×20 +27.5%
6Qassim UniversitySaudi Arabia 50.412.8%12.7×15 +212.7%
7Harbin Institute of TechnologyChina 49.622.0%3.3×24 -7.8%
8Xi'an Jiaotong UniversityChina 48.710.4%3.5×28 -0.2%
9National University of Defense TechnologyChina 48.721.8%4.8×17 +95.6%
10University of MacauChina 48.017.9%6.6×10 +145.7%

Universities only. The score blends excellence (40%), size (30%), specialisation (10%), growth (10%), international reach (10%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise.

Every institution in Numerical methods for differential equations, country by country

Is Numerical methods for differential equations research growing?

Output in 2022-2025 was 14% higher than in 2018-2021, peaking in 2025. The fastest-growing topics are Numerical methods for differential equations.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Numerical methods for differential equations as a percentage of every work published that year, 0.023% in 2024.

still counting0.050%0.10%Numerical methods for differential equations0.022% in 202519801990200020102020

Fractional works of Numerical methods for differential equations over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Numerical methods for differential equations10.7% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

Discussion

Ask a question or share a note about Numerical methods for differential equations. Posts are plain text and can be reported.