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Advanced Numerical Methods in Computational Mathematics

Advanced Numerical Methods in Computational Mathematics is a research topic within Computational Mechanics. Science Explorer counts 38k research works in it since 1950. 9.9% of them reached the world's top 10% most cited for their field and year.

This cluster of papers represents advancements in finite element methods, particularly focusing on their application to fluid-structure interaction problems. The papers cover topics such as discontinuous Galerkin methods, high-order schemes, adaptive mesh refinement, stabilized methods, multiscale modeling, preconditioners, variational methods, and PDE-constrained optimization.

  • Finite Element Methods
  • Fluid-Structure Interaction
  • Discontinuous Galerkin Methods
  • High-Order Schemes
  • Adaptive Mesh Refinement
  • Stabilized Methods
  • Multiscale Modeling
  • Preconditioners
  • Variational Methods
  • PDE-Constrained Optimization
Research works
38k
fractional, since 1950
In the world top 10%
3.8k
per year above
Top-10% rate
9.9%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+3%
the tick is no change
Largest producer
China
39% of the 10 largest, 2022-now
Advanced Numerical Methods in Computational Mathematics

Which countries lead Advanced Numerical Methods in Computational Mathematics research?

By volume, China and the United States publish the most (1.7k and 782 works in 2022-now).

By volume, 2022-now

  1. 1 China 1.7k works
  2. 2 United States 782 works
  3. 3 Germany 422 works
  4. 4 France 331 works
  5. 5 India 253 works
  6. 6 Italy 225 works
  7. 7 Russia 198 works
  8. 8 United Kingdom 152 works
  9. 9 Spain 128 works
  10. 10 Japan 95 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: 39.3%United States: 18.4%Germany: 9.9%France: 7.8%6 others listed: 24.6%39%largest
China1,676 · 39.3%United States782 · 18.4%Germany422 · 9.9%France331 · 7.8%6 others listed1,050 · 24.6%

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

Which institutions lead Advanced Numerical Methods in Computational Mathematics research?

By volume in 2022-now, Centre National de la Recherche Scientifique publishes the most Advanced Numerical Methods in Computational Mathematics research, followed by Russian Academy of Sciences and Chinese Academy of Sciences.

By volume, 2022-now

  1. 1 Centre National de la Recherche ScientifiqueFrance 133 works
  2. 2 Russian Academy of SciencesRussia 87 works
  3. 3 Chinese Academy of SciencesChina 55 works
  4. 4 Xinjiang UniversityChina 41 works
  5. 5 Xi'an Jiaotong UniversityChina 41 works
  6. 6 Shandong UniversityChina 37 works
  7. 7 Xiangtan UniversityChina 31 works
  8. 8 Helmholtz Association of German Research CentresGermany 30 works
  9. 9 Politecnico di MilanoItaly 30 works
  10. 10 Leibniz University HannoverGermany 26 works

Who are the leading researchers in Advanced Numerical Methods in Computational Mathematics?

The most-cited researchers publishing on Advanced Numerical Methods in Computational Mathematics include A. Meyer, Kai You Wang and Michael I. Jordan.

  1. 1 A. Meyer France 224k citations
  2. 2 Kai You Wang China 224k citations
  3. 3 Michael I. Jordan United States 182k 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 Advanced Numerical Methods in Computational Mathematics?

The most cited paper in Advanced Numerical Methods in Computational Mathematics 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 "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement" (6,340) and "Element‐free Galerkin methods" (5,653).

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 Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement Thomas J.R. Hughes, J. Austin Cottrell, Yuri Bazilevs · Computer Methods in Applied Mechanics and Engineering Open access Feb 2005 6,340
3 Element‐free Galerkin methods Ted Belytschko, Ye Lu, Linxia Gu · International Journal for Numerical Methods in Engineering 30 Jan 1994 5,653
4 Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations A. Brooks, Thomas J.R. Hughes · Computer Methods in Applied Mechanics and Engineering Sep 1982 5,374
5 Solution of the implicitly discretised fluid flow equations by operator-splitting Raad I. Issa · Journal of Computational Physics 1986 5,314
All 99 most cited papers, with filters and sorting

Where is Advanced Numerical Methods in Computational Mathematics research done?

The largest centres of Advanced Numerical Methods in Computational Mathematics research in 2022-now are Beijing (China), Shanghai (China), Xi'an (China) and Paris (France). 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 253 works
  2. 2 ShanghaiChina 99 works
  3. 3 Xi'anChina 97 works
  4. 4 ParisFrance 96 works
  5. 5 MoscowRussia 93 works
  6. 6 NanjingChina 88 works
  7. 7 JinanChina 82 works
  8. 8 ChangshaChina 72 works
  9. 9 Hong KongChina 54 works
  10. 10 WuhanChina 52 works

Where it is the local speciality

  1. XiangtanCN · 37.3 works15×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Advanced Numerical Methods in Computational Mathematics than the world average.

See Advanced Numerical Methods in Computational Mathematics on the map

Where is the best place to study Advanced Numerical Methods in Computational Mathematics?

Among universities, judged by research, Politecnico di Milano, Hong Kong Polytechnic University and Leibniz University Hannover 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%mean 23.45%fractional works in this field (log) →share of work in the world top 10% →Hong Kong Polytechni…Universitat Politècn…Politecnico di MilanoLeibniz University H…
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
1Politecnico di MilanoItaly 68.119.2%8.7×30 +9.7%
2Hong Kong Polytechnic UniversityChina 67.725.6%4.1×17 +45.8%
3Leibniz University HannoverGermany 66.518.7%18.8×26 +37.5%
4Universitat Politècnica de CatalunyaSpain 66.221.4%15.4×26 -29.5%
5Rice UniversityUnited States 64.022.6%12.5×13 +4.9%
6Indian Institute of Technology RoorkeeIndia 62.628.1%5.5×12 +48.5%
7Southern University of Science and TechnologyChina 61.423.3%5.2×13 +88.5%
8Chongqing Normal UniversityChina 61.035.8%12.8×8 +41.6%
9University of Milano-BicoccaItaly 60.422.3%5.9×8 +38.9%
10Politecnico di TorinoItaly 59.817.5%8.9×19 +87.2%

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 Advanced Numerical Methods in Computational Mathematics, country by country

Is Advanced Numerical Methods in Computational Mathematics research growing?

Output in 2022-2025 was 3% higher than in 2018-2021, peaking in 2014. The fastest-growing topics are Advanced Numerical Methods in Computational Mathematics.

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 Advanced Numerical Methods in Computational Mathematics as a percentage of every work published that year, 0.023% in 2024.

still counting0.020%0.040%0.060%Advanced Numerical Methods in Computational Mathematics0.022% in 202519801990200020102020

Fractional works of Advanced Numerical Methods in Computational Mathematics 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%Advanced Numerical Methods in Computational Mathematics10.0% 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.

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