Ao carregar a pagina está vindo em branco, depois verifiquei que está havendo um erro no modo de autenticação no cliente mysql:

código do produtos.js:

module.exports = function(app){
  app.get('/produtos',function(req,res){
    var mysql = require ('mysql');
    var connection = mysql.createConnection({
      host : 'localhost',
      user : 'root',
      password : 'root',
      database : 'casadocodigo_nodejs'
    });
    connection.query('select * from livros', function(err,results){
      if(err){console.log(err);}
      console.log(results);
      res.send(results);
    });
    connection.end();
  });
}

Erro obtido na query:

{ Error: ER_NOT_SUPPORTED_AUTH_MODE: Client does not support authentication prot
ocol requested by server; consider upgrading MySQL client
    at Handshake.Sequence._packetToError (C:\Users\Sergio\estudoNode\casadocodig
o\node_modules\mysql\lib\protocol\sequences\Sequence.js:52:14)
    at Handshake.ErrorPacket (C:\Users\Sergio\estudoNode\casadocodigo\node_modul
es\mysql\lib\protocol\sequences\Handshake.js:130:18)
    at Protocol._parsePacket (C:\Users\Sergio\estudoNode\casadocodigo\node_modul
es\mysql\lib\protocol\Protocol.js:279:23)
    at Parser.write (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\mysql\
lib\protocol\Parser.js:76:12)
    at Protocol.write (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\mysq
l\lib\protocol\Protocol.js:39:16)
    at Socket.<anonymous> (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\
mysql\lib\Connection.js:103:28)
    at emitOne (events.js:116:13)
    at Socket.emit (events.js:211:7)
    at addChunk (_stream_readable.js:263:12)
    at readableAddChunk (_stream_readable.js:250:11)
    --------------------
    at Protocol._enqueue (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\m
ysql\lib\protocol\Protocol.js:145:48)
    at Protocol.handshake (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\
mysql\lib\protocol\Protocol.js:52:23)
    at Connection.connect (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\
mysql\lib\Connection.js:130:18)
    at Connection._implyConnect (C:\Users\Sergio\estudoNode\casadocodigo\node_mo
dules\mysql\lib\Connection.js:461:10)
    at Connection.query (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\my
sql\lib\Connection.js:206:8)
    at C:\Users\Sergio\estudoNode\casadocodigo\app\routes\produtos.js:10:16
    at Layer.handle [as handle_request] (C:\Users\Sergio\estudoNode\casadocodigo
\node_modules\express\lib\router\layer.js:95:5)
    at next (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\express\lib\ro
uter\route.js:137:13)
    at Route.dispatch (C:\Users\Sergio\estudoNode\casadocodigo\node_modules\expr
ess\lib\router\route.js:112:3)
    at Layer.handle [as handle_request] (C:\Users\Sergio\estudoNode\casadocodigo
\node_modules\express\lib\router\layer.js:95:5)
  code: 'ER_NOT_SUPPORTED_AUTH_MODE',
  errno: 1251,
  sqlMessage: 'Client does not support authentication protocol requested by serv
er; consider upgrading MySQL client',
  sqlState: '08004',
  fatal: true }

já tentei executar novamenteo npm install mysql --save e já utilizei npm i npm para atualizar o modulo do MySql, o que mais pode ser? Alguem sabe como resolver?