<role rolename="manager"/>
<role rolename="admin"/>
<user username="user" password="password" roles="manager,admin"/>

<Realm className="org.apache.catalina.realm.UserDatabaseRealm"
       resourceName="UserDatabase">
   <CredentialHandler className="org.apache.catalina.realm.SecretKeyCredentialHandler"
                      algorithm="PBKDF2WithHmacSHA512"
                      keyLength="256"
   />
</Realm>

$CATALINA_HOME/bin/digest.sh -a "PBKDF2WithHmacSHA512" -i 100000 -s 16 -k 256 -h "org.apache.catalina.realm.SecretKeyCredentialHandler" "YOUR_PASSWORD"

<!-- for password “admin” -->
<user username="tomcat" password="8c6976e5b5410415bde908bd4dee15dfb167a9c873fc4bb8a81f6f2ab448a918" roles="manager-gui,manager,admin"></user>

<catalina_home>/webapps/managerWEB-INFweb.xml

    <login-config>
        <auth-method>DIGEST</auth-method>
        <realm-name>UserDatabase</realm-name>
    </login-config>

Full context:
  <servlet>
    <servlet-name>HTMLManager</servlet-name>
    <servlet-class>org.apache.catalina.manager.HTMLManagerServlet</servlet-class>
    <init-param>
      <param-name>debug</param-name>
      <param-value>2</param-value>
    </init-param>

… SNIPPED_FOR_BREVITY ...

    <login-config>
    <auth-method>DIGEST</auth-method>
    <realm-name>UserDatabase</realm-name>
  </login-config>

    <multipart-config>
      <!-- 50MB max -->
      <max-file-size>52428800</max-file-size>
      <max-request-size>52428800</max-request-size>
      <file-size-threshold>0</file-size-threshold>
    </multipart-config>
  </servlet>

<login-config>
    <auth-method>DIGEST</auth-method>
    <realm-name>UserDatabase</realm-name>
</login-config>